Computational Fluid Dynamics A Practical Approach, 2nd Edition Jiyuan Tu, Guan Heng Yeoh, Chaoqun Li

SOLUTION MANUAL COMPLETE UPDATE

REVIEW QUESTIONS 1.1 From which industry has CFD emerged from?

The emergence of CFD came from the high-technology engineering industries of aerospace and astronautics. It’s application has since been adapted for use in many engineering applications.

1.2 Which three disciplines CFD is derived from?

CFD integrates three disciplines: • Fluid Mechanics – this involves the study of fluids at rest and in particular fluids in motion. This includes the fluid flow behaviour and flow processes such as heat and mass transfer, and chemical reactions.

• Mathematics – this involves the physics of fluid motion governed by a set of mathematical equations (partial differential equations).

• Computer Science – this involves computational techniques that are used to solve complex mathematical equations computationally.

1.3

Traditionally, CFD has been used to solve aerospace and automotive engineering applications such as drag and lift for airplanes and cars, what examples can you think of where CFD is being used within non-traditional fluid engineering applications?

Some examples of non-traditional engineering applications include: • bio-medical engineering that deals with blood flow and respiratory flow; • sports performance such as swimming techniques, aerodynamic cycling, and yachting/rowing boat and oar design; • renewable power generation through wind power blade designs and heat transfer of solar power production; • civil and environmental engineering that deals with air and water flow around infrastructures (oil rigs or high skyscrapers), coastal water movements, wind and fire simulations.

1.4 What are some of the advantages of using CFD? Some advantages of CFD include: • Through computational science, specific terms in the governing equations can be studied in a more detailed fashion which is difficult through analytical mathematics.

• CFD results can be used to complement experimental and analytical approaches which are a cost-effective means of simulating real fluid flows.

• The ability of simulating flow conditions that are not reproducible in experimental tests found in geophysical and biological fluid dynamics, such as nuclear accident scenarios or scenarios that are too huge or too remote to be simulated experimentally (e.g. Indonesian Tsunami of 2004) • CFD results are detailed and provides comprehensive visual information when compared to analytical and experimental methods.

•

CFD simulations can speed up experiments through virtual testing of alternate evaluated over a range of dimensionless parameters that may include the Reynolds number, Mach number, Rayleigh number and flow orientation.

.

1.5 What are the limitations and disadvantages of using CFD? The main limitation of CFD exists in the solution of the mathematical equations that govern fluid motion by numerical methods. Numerical errors occurs which creates differences between the computed results and reality.

Visualization of numerical solutions may present attractive images that appear to provide a sense of realism but there is a danger that it is an erroneous solution due to numerical errors. Hence, CFD simulations require the user to have some knowledge of the three facets of CFD (computational science, mathematics and fluid dynamics) in order to properly analyze and make critical judgment on the computed results.

1.6

What CFD measurements can be obtained to assist the designs for safety and comfort in passenger airplanes?

Data regarding the lift and drag, and the impact of trailing vortices for the safe operation of aircraft taking-off and landing on the runway can be obtained to assist designs for safety issues. The internal cabin ventilation, heating and cooling system can be simulated and measured to enhance the passenger and crew comfort.

1.7

How is CFD being used as a research tool, a design tool and an educational tool in academic fields, such as Thermal-Fluids?

CFD can be employed as: • a research tool to perform numerical experiments. Comparative visualization of the numerical calculations with experimental results allows a deeper investigation into the observed flow structures and physical aspects of a flow field, similar to a real laboratory experiment, e.g. obtaining heat transfer correlations; • a design tool, parameters can be isolated and tested to effects of different designs, e.g. design of heat exchangers; • an educational tool that provides visualization capability enhancing the students’ intuition of the flow behavior, e.g. visualization of convective flows.

1.8

How can CFD be applied and used to improve cost-effective design procedures in the automotive industry?

CFD simulations have the ability to shorten design cycles through optimization of existing engineering components and systems to improve energy efficiency and meet strict standards and specifications. CFD can be used to determine the small changes in pre-existing designs such as the effects of local geometry changes on the aerodynamic forces. This procedure reduces the dependence on time-consuming expensive clay models and wind-tunnel experiments and delivers quicker design turn-around.

1.9

The bio-medical science field is turning to CFD to resolve flows within human airway and vascular systems. What advantages does CFD hold over experiments in obtaining these numerical results?

Experimental testing for the human airway and vascular system is intrusive, difficult and expensive while ethical issues have to be considered. CFD can provide flow simulations of the human body, and also virtual prototyping to recommend the best design for surgical reconstructions such as carotid endarterectomy. This can lower the chances of post-operative complications, assist in developing better surgical procedures and deliver a good understanding of biological processes as well as more efficient and less destructive medical equipment such as blood pumps.

1.10

What details can CFD capture in the simulation of hydro-cyclones, a process commonly used in the minerals industry?

CFD can capture the separation of the solid particles from the liquid due to the imparted centrifugal forces inside the hydro-cyclone. Individual particles can be tracked throughout the system through different multiphase (here liquid and solid particles are two different phases) flow approaches.

1.11 How is CFD being used in the civil and environmental industry?

Infrastructure in the civil industry can use CFD to test the flow of fluids, especially for large water tank and pipe flow in dam designs. CFD has been used to predict the pollutant plume dispersed from a cooling tower subject to wind condition which helps to ascertain the effects of the exhaust plume on the surrounding structures. Also architectural structures exposed to environmental elements can be tested (e.g. the Itsukushima Torii (Gate) ???.

1.12

What two applications see CFD being used in the power generation industry? What kind of data is collected and how is this useful in increasing the efficiency in power generation?

Turbine blade designs for wind generation are tested through CFD. The amount of torque created under varying wind conditions and the positioning of the turbines throughout a complex terrain with steep inclines, is obtained to maximise power generation. Also simulations of combusting fluidized coal beds as well as boiler operation can provide the necessary data regarding, fuel consumption, pollutant emissions, slagging and degradation of the tubes, thus improving the efficiency of power generated through coal.

1.13

How can CFD influence the way swimmers in improving their swimming strokes?

CFD has helped to “design” the optimum stroke to achieve peak propulsive performance for elite swimmers. Investigations evaluated the flow around the hand and forearm of a swimmer during the propulsion phases of the freestyle and butterfly strokes. Steady-state lift and drag forces for evaluated at different angles of attack ranging from -15 to 195 degree to find the optimum stroke.

1.14 What competitive edge can CFD give to a cycling team?

The design of the bikes’ forks and handlebar arrangement were evaluated to see the effects of the rider’s positioning on the bike on the drag that is caused

by the overall shape of the cyclist. Additionally a streamlined design for the aerodynamic helmet was tested and created to accommodate different head styles achieving the ultimate cycling efficiency.

1.15

In the future, to what extent will CFD be involved within the product development process in manufacturing?

In the future, CFD will be an integral part in many manufacturing processes.

Its main advantage is the ability to cut time and costs through virtual modeling instead of prototyping. By performing time-dependent simulations every possible design aspect and engineering judgment can be consistently exercised on the spot through real-time assessments on proposed customized design simulations in selecting the most optimum vehicle.

1.16 What is the future of CFD? Through the increase in computing power, CFD techniques are infiltrating more industries and different areas of study. Complex fluid mechanics problems such as jet flames, buoyant fires, multi-phase and/or multicomponent flows are now being progressively applied especially through the availability of multi-purpose commercial CFD computer programs.

Advancements will be made to more accurate turbulence models such as the Large Eddy Simulation (LES) which will become the preferred methodology for many turbulence investigations of fundamental fluid dynamics problems.

Also more unsteady simulations with large complex geometries, such as car aerodynamics with dynamic movements like cornering and turning will be feasible in the future.

REVIEW QUESTIONS – CHAPTER 2 2.1

How are commercial codes allowing CFD analysis to be carried out with ease for the novice user?

CFD commercial codes employ more user-friendly environments and provide functionality for a variety of applications. Without any prior basic CFD knowledge, a first-time user can perform the necessary operations that involve setting up a fluid problem, solving the numerical problem and managing some graphical representation of the results attained.

2.2 What are the main elements involved in a complete CFD analysis?

The main elements involved are: the pre-processing step that involves creating the computational domain and geometry from the physical situation; the solver which implements the numerical techniques and applies it across the computational domain; the post-process step, which involves presenting the data in graphical format for analysis.

2.3

Why is it important to correctly define the computational domain for the fluid flow problem? Give an example of this.

The computational domain must represent the physical domain correctly so that the results will reflect the true nature of the real case. The following is an example of the many ways the physical domain of a flow passing two cylinders can be converted into a computational domain. Note that the computational domain requires boundary definitions to enclose the domain.

Physical domain

Flow Direction

Two cylinders

Computational Domain Flow into domain through open boundary as inflow boundary

Or flow out of domain as outflow boundary condition

Open boundary Inflow boundary

Two

Outflow boundary

Velocity = 0; Temperature or Heat Flux given

Flow Direction Open boundary Density; Velocity; Temperature given

2.4

What is the consequence of using a very fine mesh (i.e. a very large number of cells) compared with using a coarse mesh (i.e. a small number of cells)?

A very fine mesh uses a large number of cells, which means that more calculations are required to iterate the governing equations to resolve for all the cells. Therefore using a coarser mesh will lead to a shorter computational simulation time. However a coarse mesh will exhibit large cell sizes which lead to less accurate results.

2.5

What is the main difference between a structured and unstructured mesh and when are they applied to physical domains?

For relatively simple geometries such as a square box, structured cells that generally comprise a regular distribution of rectangular cells is a better option. For more complex geometries such as the human body, structure cells are difficult to apply and the use of unstructured cells is appropriate.

2.6 What types of boundary conditions can be imposed on the computational domain?

The type of boundary condition that needs to be imposed on the computational domain is dependent on the physical model. The following are some of the available conditions: Inflow - pressure, velocity, mass-flow Outflow - pressure, velocity, mass-flow Open - symmetry, axis, cyclic, wall

2.7

What type of boundary can be used for a computational boundary that represents an open physical boundary?

The most common boundary conditions for an open physical boundary is symmetry, axis and cyclic.

2.8

What advantages can a symmetry boundary condition and a cyclic boundary condition provide and when can they be applied?

Symmetry or cyclic boundary conditions can be applied on repeating geometries, where a single computational domain is used to represent the entire domain through symmetry or cycles.

Physical Domain

Flow Direction

Computational Domain

The simplified computational domain is now smaller and a very fine mesh resolution can be applied. The computations are now applied to one representative domain rather than the entire physical domain that would take a much longer time.

2.9 What is the main purpose of a CFD solver?

The purpose of the CFD solver is to set up numerical algorithms and to obtain a solution for the governing equations. The solver consists of applying different models and numerical schemes to control the accuracy and stability of the simulation followed by computationally solving the problem.

2.10 What are the advantages of providing intelligent values for the initial solution?

Although initial conditions can be applied arbitrarily, an intelligent initial value that is close to the final solution will quicken the iterative procedure whereby the solution will converge in shorter computational time. If, however, the initial values are far away from reality, the computations will take longer for the solution to converge and may even lead to the iterative procedure misbehaving and possibly “blowingup” or diverging. Normally an intelligent initial guess is based on the user’s experience.

2.11 What is an iteration process and how is it performed?

The iteration process involves the solving of the algebraic set of equations leading to a set of solved values for all the variables involved. Some well known iterative solvers are Algebraic Multi Grid

(AMG) algorithm, conjugate gradient methods, Strongly Implicit Procedure (SIP) of Stone’s method and TDMA line-by-line procedure. The goal of the iteration is to reduce the magnitude of the differences in the solution within each iteration step. The iteration process stops when the differences, called residuals, reach a certain limit or criteria. More descriptions of iterative solvers are provided

in Chapter 4.

2.12 What does the convergence criterion control?

A solution is considered to have converged when the differences in the solution variables, also called residuals or imbalances, is reduced through each iteration step until an acceptable level is reached. This acceptable level is called the convergence criteria. More details of residuals are discussed in Chapters 4

and 5.

2.13 What is the main purpose of the post-processing stage?

The purpose of the post-processing stage is to translate, interpret and analyse the raw numerical data into graphical formats that are visually appealing and that convey the results in the correct manner.

2.14

What are the advantages of using X-Y plots? Give examples of what CFD results X-Y plots can capture.

X-Y plots are usually two-dimensional graphs that present the variation of one dependent variable against another independent variable. They are the most precise and quantitative way to present the numerical data since they are easily identifiable. The reader can interpret the results easily without resorting to any mental or arithmetic interpolation.

Examples of X-Y plots may involve line profiles of velocity, pressure, shear stress, temperature and skin-friction coefficient.

Ave Wall Shear Stress

Ave Cell Temperature

0.04

0.03 31.8 0.02

31.5 0.01

0

Ave. Cell Temperature (Celcius)

Shear Stress (1/s)

32.1

31.2 0

400000

800000

1200000

Mesh Resolution

Example of two x-y plots on a single chart 2.15 Why are contour plots best used to display the distribution of a variable?

Contour plots provide a global description of the fluid flow encapsulated in one view. A contour line

(also known as isoline) is a line that indicates a property which is constant along that line.

Generally the numerical value of a variable from one contour line to an adjacent contour line is also held constant and it is the distance between each isoline that is useful in showing the distribution of the variable over the geometry. For example the below velocity magnitude contour plot of a nasal cavity cross-section shows a little distribution at the tips of the geometry where the lines are widely spaced apart, while in the middle regions where the lines are close to each other, there is a large greatest round the edges.

widely spaced isolines showing little distribution closely spaced isolines showing greater distribution

Velocity magnitude contour plot of a nasal cavity cross-section 2.16 What is the meaning of a streamline? What advantages do they have over other plot types?

A streamline is a path that is traced out if a massless particles moves with the flow. In other words any point along the streamline represents the parallel line to the mean velocity vector at that point. The streamline plot can reveal important features that could be obscured in some isolated flow regimes when using other types of plots. For example the below figure shows a streamline plot of a flow around a cylinder, which shows the wake-developing vortices behind the cylinder which may be obscured in other plots such as contours or x-y plots.

Streamline plot of a flow over a cylinder showing wake-developing vortices

REVIEW QUESTIONS - CHAPTER 3 3.1

Simplify the general continuity equation:

∂ρ ∂t

+

∂ ( ρu ) ∂x

+

∂ ( ρv ) ∂y

+

∂ ( ρ w) ∂z

= 0 , for a two-

dimensional constant density case.

The two dimensional continuity equation with a constant density can be simplified to: ∂ρ ∂ ( ρ u ) ∂ ( ρv ) ∂ ( ρ w ) + + + =0 ∂t ∂x ∂y ∂z The equation can be rearranged through the chain rule as:

⎛ ∂u ∂v ⎞ ∂ρ ∂ρ ∂ρ ⎟⎟ = 0 +u +v + ρ ⎜⎜ + ∂t ∂x ∂y ⎝ ∂x ∂y ⎠ since ρ is constant then all the derivatives of ρ are zero. i.e.

∂ρ ∂ρ ∂ρ = = =0 ∂t ∂x ∂y

and

∂u ∂v + =0 ∂x ∂y

3.2

In a converging nozzle, the flow accelerates due to the narrowing geometry. Discuss the ∂v ∂u changes in the velocity gradients and during the flow (assume a constant density). ∂y ∂x ∂u ∂v + = 0 . For a horizontal flow in a converging nozzle, The constant density continuity equation is

∂x

∂y

the fluid experiences an acceleration as its cross-sectional area decreases. The velocity at the position x increases to a greater velocity at x + Δx. This means that the velocity gradient in the x direction is positive, ∂u/∂x > 0. The vertical velocity component, v therefore gets smaller, since more fluid is physically moving the control volume than entering along the x direction, there should be more fluid entering than leaving along the y direction. Here, ∂v/∂y < 0 and the velocity at the surface y + Δy is less than the velocity at the surface y, i.e. v(y + Δy) < v(y).

3.3 What is Newton’s second law of motion?

Newton’s second law of motion states that the net sum of all the forces that is acting on an object, equals to the product between its mass and acceleration of the element.

Example:

FT = 15N

Fy = 12N

Fx = 9N Friction =2N

mass = 5kg

Weight = 49N

Normal = 37N

∑ = ma = 9 − 2 = 7 N Fx

x

ax = 1.4 m/s2

∑ = ma = 12 + 37 − 49 = 0N Fy

y

ay = 0 m/s2

3.4 Write a force balance equation for all the forces acting on a differential control volume.

There are two types of forces acting on a fluid control volume which are: surface forces - pressure forces - shear forces body forces - gravity force - electromagnetic force - other forces (e.g. Coriolis force, centrifugal force) It is common to add the surface forces as separate terms in the force balance equation while including the effects of body terms as source terms denoted by S. The x-component of the surface forces on a differential control volume is given below: where the σ xx is the normal stress (sometimes given as Pressure)

τ xy ,τ xz are the tangential stresses

y, v

x, u

y + Δy

∂τ yx ⎤ ⎡ Δy ⎥ ΔxΔz ⎢τ yx + ∂y ⎣ ⎦

τzx ΔxΔy

z, w

σxx ΔyΔz y

∂τ zx ⎤ ⎡ ⎢τ zx + ∂z Δz ⎥ ΔxΔy ⎣ ⎦

z + Δz

x z

x + Δx

∂σ xx ⎤ ⎡ ⎢σ xx + ∂x Δx ⎥ ΔyΔz ⎣ ⎦

τyx ΔxΔz Surface Forces

From Newton’s second law of motion the net sum of the surface and body forces is balanced by the net rate of increase in the momentum. For the x-momentum per unit volume this is given as:

ρ

Du = ∑ Forces Dt

where

∑

⎞ ⎛ ∂τ yx ∂σ xx ⎛ ⎞ ⎛ ∂τ zx ⎞ ⎜⎜ = σ − σ − Δ x y z z τ τ x y Δ Δ + Δ + − Δ Δ + Δy + τ yx − τ yx ⎟⎟ΔxΔz + S x ⎜ ⎟ ⎜ ⎟ xx xx zx zx Forces ∂x ⎝ ⎠ ⎝ ∂z ⎠ ⎠ ⎝ ∂y ⎛ ∂σ xx ∂τ zx ∂τ yx ⎞ ⎟⎟ΔxΔyΔz + S x = ⎜⎜ − + + x z y ∂ ∂ ∂ ⎠ ⎝

3.5

For the momentum of a fluid property in the x direction discuss how the local acceleration ∂u ∂u ∂u +v and the advection terms u contribute to the overall transport of the fluid. ∂x ∂y ∂t The momentum of a fluid property accelerates due to two effects: i) the local acceleration with respect to time and ii) the local acceleration with respect to its surroundings or its space. At any point in a flow, the local acceleration with respect to time is defined as

∂u . This means that the change in velocity is ∂t

dependent on time only. Additionally, the acceleration due to velocity gradients in the flow at any point is given by u

∂v ∂u , v . This advection term describes how the velocity is accelerating due to local velocity dx ∂y

gradients. The local time-dependent acceleration and the acceleration from advection both contribute to the net acceleration of the fluid at a given point.

3.6

A simplified one-dimensional inviscid, incompressible, laminar flow is defined by the ∂u ∂u 1 ∂p +u =− following momentum equation in the x direction: . Name each term and ∂t ∂x ρ ∂x discuss their contribution to the flow.

∂u dt ∂u u ∂x

is the x-component of the local acceleration with respect to time which describes how the flow’s acceleration changes with time. is the x-component advection term which is the local acceleration with respect to space. It describes how the acceleration changes due to the velocity gradient.

1 ∂p is the pressure gradient which is the result of the net surface force acting on the flow in the x− ρ ∂x direction 3.7

The momentum of a fluid in the y direction is given by the following equation: 1 ∂p ∂v ∂v ∂v ∂ 2v ∂ 2v +u +v =− +ν +ν − g . Discuss the forces that act to transport the ∂t ∂x ∂y ρ ∂y ∂x 2 ∂y 2 fluid.

−

1 ∂p is the pressure gradient which is the result of the net surface force acting on the flow in the yρ ∂y

ν

∂ v ∂ 2v , ν ∂2x ∂2 y

direction 2

are the viscous diffusion terms

−g

is the gravity term that is acting downwards The resultant net force of the terms on the right hand side of the equation results in the acceleration of the fluid. The acceleration is equal to the terms on the left hand side of the equation which is made up of the local acceleration,

3.8

∂v ∂v ∂v and the advective acceleration, u +v ∂x ∂y ∂t

What are the differences between the momentum equation in question 3.7 and the following ⎛ ∂ 2u ∂ 2 u ⎞ ∂u ∂u ∂p + + ρv =− +μ⎜ momentum equation: ρ u ⎟. ⎜ ∂x 2 ∂y 2 ⎟ ∂x ∂y ∂x ⎝ ⎠

There are two physical differences between the momentum equation in Question 3.7 and the above momentum equation. One is the absence of the gravity term, which implies that the influence of gravity is negligible. The second difference is the momentum equation above is for the x-momentum component whereas the equation in Question 3.7 is the y-momentum equation. The fact that the appearance of the equations are different is due to the algebraic manipulation of μ = υρ .

3.9

The hydrodynamic length for a channel flow shown below is equal to Le when air is used (uin = 0.03 m/s, μ1 = 1.65 × 10-5 kg/m ⋅ s, and ρ = 1.2 kg/m3). To obtain the same hydrodynamic length for water, what inlet velocity is required (μ1 = 1.003 × 10-3 kg/m ⋅ s, ρ = 1000 kg/m3).

Wall (Stationary Plate)

Uniform Velocity

y Inlet

H

x

Outlet

Wall (Stationary Plate) LE

U1 = 0.03m/s µ1 = 1.65x 10-5 kg/m.s ρ1 = 1.2 kg/m3 Re1 =

ρUD 1.2(0.03) = D μ 1.65 x 10 −5

= 2182 D

U2 = ? m/s µ2 = 1.003 x 10-3 kg/m.s ρ2 = 1000 kg/m3 Re 2 =

ρUD 1000 = UD μ 1.003 x 10 −3

= 997009UD

To achieve the same hydrodynamic features the Reynolds number must equal.

Re1 = Re 2 2182 D = 997009UD ⇒ 2182 = 997009U

U = 0.0022 m/s

3.10 The Reynolds number is a ratio of two fluid properties. What are they?

The Reynolds number is a ratio of the fluid momentum (inertia) to the fluid’s diffusion (viscosity).

Re =

inertia ρUD UD = = viscosity μ υ

3.11

If the Reynolds is very high (Re » 10,000), what does this suggest? If it is very low (Re « 100), what does this imply?

A high Reynolds number implies that the inertial forces are large relative to the viscous forces and therefore the viscous forces cannot prevent the random and rapid fluctuations of the fluid. The inertia forces are proportional to the fluid density, fluid velocity and a hydrodynamic length. At a low Reynolds number, however, the viscous forces are large enough to overcome the inertial forces which smoothes the flow. The Reynolds number is commonly used to determine whether a flow is turbulent or laminar. For the above cases of a high and low Reynolds number, the flow is considered turbulent and laminar respectively.

3.12 Explain the first law of thermodynamics.

The first law of thermodynamics states that the net change (increase or decrease) in the total energy of a system is equal to the difference between the total energy entering and the total energy leaving the system. e.g.

⎛ Total energy ⎞ ⎛ Total energy ⎞ ⎛ Net change in the ⎞ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ entering the ⎟ − ⎜ leaving the ⎟ = ⎜ total energy of ⎟ ⎟ ⎟ ⎜ the system ⎟ ⎜ system ⎜ system ⎠ ⎠ ⎝ ⎠ ⎝ ⎝ This is known as the conservation of energy principle which means that energy can neither be created nor destroyed; it can only change forms. This implies that during a process all forms of energy have to be accounted for. Normally in CFD, energy can come in the form of heat, work and mass flow. Therefore the conservation of energy balance equation usually involves:

E in − E out + E gen = ΔE system Net energy transfer by heat, work and mass

3.13

Internal energy generation

Change in the total energy of the system

Name the sources of energy that contribute to the energy equation.

The energy equation considers the heat energy from the fluid properties such as the temperature, and molecular viscosity. These quantities are transported across the flow which contributes to the total energy of the system. These transported quantities are defined by the following terms: ∂T ∂ (uT ) ∂(vT ) ∂ (wT ) + + + Æ local and advective acceleration of temperature ∂t ∂x ∂y ∂z

ν T ⎞ ∂T ⎤ ∂ ⎡⎛ ν ⎟ ⎢⎜⎜ + ⎥ ∂xi ⎣⎝ Pr PrT ⎟⎠ ∂xi ⎦

Æ

viscous diffusion of the temperature, where the subscript i

represents 1,2,3 for the three dimensions of x,y,z.

3.14

Write an equation for the energy balance using the sources of energy defined in question 3.13 for the first law of thermodynamics.

ν T ⎞ ∂T ⎤ ∂ ⎡⎛ ν ν T ⎞ ∂T ⎤ ∂ ⎡⎛ ν ν T ⎞ ∂T ⎤ ∂T ∂ (uT ) ∂ (vT ) ∂ (wT ) ∂ ⎡⎛ ν ⎟ ⎥ + ⎢⎜ + ⎟ ⎥ + ⎢⎜ + ⎟ ⎥ + ST + + + = ⎢⎜⎜ + ∂t ∂x ∂y ∂z ∂x ⎣⎝ Pr PrT ⎟⎠ ∂x ⎦ ∂y ⎣⎜⎝ Pr PrT ⎟⎠ ∂y ⎦ ∂z ⎣⎜⎝ Pr PrT ⎟⎠ ∂z ⎦ 3.15

Apply Fourier’s law of heat conduction to obtain the heat flux in the x-direction.

The heat flux in the x-direction is: ∂q − x ∂x where ∂T q x = −k ∂x By substitution we get ∂ 2T ∂ ⎡ ∂T ⎤ k = k ∂x ⎢⎣ ∂x ⎥⎦ ∂x 2 3.16

Write the equation that defines the substantial derivative for the transport of temperature in terms of the local acceleration derivative and the advection derivative of temperature?

∂T ∂T ∂T DT ∂T = +u +v +w ∂t ∂x ∂y Dt ∂z 3.17

If a car travels across a warmer environment, the car body will experience a rise in temperature. Is this an example of the local acceleration derivative or advection derivative at work?

This is an example of the advection derivative at work where the temperature itself varies through the

environment (space) that the car moves across. A local acceleration derivative may occur if there is some rain that suddenly drops the temperature.

3.18

In what type of situation can you simplify the general 2D energy equation: k ∂ 2T k ∂ 2T ∂T ∂T ∂T to reach the well-known Laplace’s equation: +u +v = + ∂t ∂x ∂y ρ C p ∂x 2 ρ C p ∂y 2 ∂ 2T ∂ 2T + = 0. ∂x 2 ∂y 2 The general 2D Laplace’s equation has been simplified due to the following: • it is a steady flow, ∴ the time derivative becomes zero • there is no fluid motion ∴ the terms on the left hand side becomes zero 3.19

Obtain the general analytical solution for Laplace’s equation for a one-dimensional case.

d 2T ∂T k ∂ 2T ∂T Î =0 +u = ∂t ∂x ρC p ∂x 2 dx 2 The energy equation in the x-direction has been simplified to Laplace’s equation

d 2T = 0 where the dx 2

flow is steady and there is no fluid flow but rather a solid state problem. Therefore the simplification is:

=0 =0 =0 ∂T k ∂ 2T ∂T +u = ∂t ∂x ρC p ∂x 2 and d 2T =0 dx 2

Integrating the differential equation once with respect to x yields ∂T = C1 ∂x where C1 is an arbitrary constant. Subsequent integration of the above equation yields T (x ) = C1 x + C2 which is the analytical expression of the differential equation. The general solution contains two unknown constants C1 and C2. We need two equations which can be determined through the boundary conditions imposed on the left and right surfaces of the solid slab.

3.20 The Prandtl number is a ratio of two fluid properties. What are they?

The Prandtl number is a dimensionless parameter defined by the ratio of the molecular diffusivities of momentum and heat.

Pr =

molecular diffusivity of momentum μC p = molecular diffusivity of heat k

3.21 Fluids such as oils have a high Prandtl number (Pr » 1). What does this suggest?

Prandtl numbers greater than 1 indicates that heat diffuses slowly relative to momentum. This means that the development of the thermal boundary layer is slower and its thickness at a given point will be thinner than the velocity boundary layer.

3.22 What is the significance of the Prandtl number equalling to 1 in terms of entry lengths.

When the Prandtl number is equal to 1, the heat and momentum diffuses at the same rate. This means the development of the boundary layer occurs at the same rate and hence the entry lengths of the fluid will be the same.

3.23 What is the energy cascade process in turbulence?

The larger eddies are dominated by inertia forces and hence effectively inviscid. The mean flow provides the energy by stretching and deforming the larger eddies. The smaller eddies are further stretched by the larger eddies. Finally the smallest eddies dissipate the energy which is converted into thermal internal energy. This transfer of kinetic energy that is passed down from larger eddies to progressively smaller and smaller eddies in a cascading manner is called the energy cascade process.

3.24

Why do large eddies tend to be anisotropic in nature? Why are small scaled eddies isotropic in nature?

The structure of larger eddies which are dominated by inertial forces and inviscid, tend to be anisotropic because of their dependence on the mean flow for its energy. The strong interaction with the mean flow causes the eddies to stretch in all directions. The structure of smaller eddies, however, are isotropic because of its diffusive nature where viscous forces are dominant. The action of viscosity will smear out directionality at these smaller scales.

3.25

The use of Direct Numerical Simulation (DNS) remains a problem for engineering applications. Why? Direct Numerical Simulation is an exact simulation for turbulent flows which accounts all scales of turbulence, even the smallest and fastest occurring eddies. This means that the mesh generated must be fine enough to capture the small length scales of turbulence and the time steps have to be very short to capture the small time scales of turbulence, which can be unrealistic for engineering applications. As an example, a typical flow domain with a cross-sectional area of 0.1 m by 0.1 m with a high Reynolds number turbulent flow might contain eddies down to 10 or 100 μm in size. In order to describe the flow processes at all length scales, a mesh of 109 to 1012 grid points is required. Also, the fastest events can take place with a frequency of the order 10 kHz, which we would need to discretise time into steps of about 100 μs.

3.26

Which profile as indicated below is the laminar or turbulent velocity profile?

1.6

a

1.4 1.2

b

u /uin

1 0.8 0.6 0.4 0.2 0 0

0.02

0.04

0.06

0.08

0.1

Height, H

Profile A is a laminar velocity profile as the central velocity is greater. The viscous forces are dominant which prevents the momentum from spreading evenly. On the other hand, profile B is a turbulent velocity profile which shows that the inertial forces are dominant and the momentum is spread out more evenly as displayed by the flatter and more even velocity distribution.

3.27

From question 3.26, describe the shape of the turbulent velocity profile? Explain why the profile is so different from the laminar profile?

The turbulent velocity profile is flatter and fuller in shape with a sharp gradient near the wall boundaries.

This turbulent profile is different from the laminar velocity profile due to the relative strengths of the inertial forces to the viscous forces as given by the Reynolds number. Turbulent profiles have dominant inertial forces while laminar profiles have dominant viscous forces.

3.28

For

an

incompressible flow, the non-dimensional transport equation can be ∂u 1 ∂ 2u . Provide an explanation why for a laminar flow, the diffusion simplified as u = ∂x Re ∂x 2 term (the right-hand side of the equation) will be more dominant than the convective term (the left-hand side of the equation). At a highly turbulent flow, the situation is opposite, why?

The diffusion term on the right is inversely proportional to the Reynolds number. Laminar flows have

1 → larger number) producing a large coefficient. Conversely if the flow is Re 1 turbulent where the Reynolds number is high and therefore → smaller number, the coefficient for Re small Reynolds numbers (

the diffusion term becomes smaller.

3.29

Indicate the terms in the following general equation: u

∂φ

=Γ

∂ 2φ + Sφ . ∂x 2

∂x The above equation involves advection and diffusion (more commonly known as convection-diffusion) applied to a general variable φ .

∂φ = advection or convection term applied to the variable φ ∂x ∂ 2φ Γ 2 = diffusion term with the diffusion variable, Γ applied to the variable φ ∂x

u

= relevant source terms of the variable φ

Sφ

The equation in question 3.29 represents a transport process for the property φ. Under what circumstances, can this equation be applied?

The diffusion-advection equation in question 3.29 applies to a one-dimensional case that is under a steady state solution.

3.30

3.31 Spot the errors in the equation below. (Hint: It is a 3D x-momentum equation) ∂u ⎤ ∂u ∂ ( uu ) ∂ (v u ) ∂ ( uu ) ∂ ⎡ ∂u ⎤ ∂ ⎡ ∂u ⎤ ∂ ⎡ + + + = + ⎢ ν +νT + ⎢ ν +νT ν +νT ⎢ ⎥ ⎥ ∂t ∂x ∂x ∂z ∂x ⎣ ∂x ⎦ ∂y ⎣ ∂x ⎦ ∂z ⎣ ∂x ⎦⎥

(

)

⎛

1 ∂p

⎝

ρ ∂x

+ ⎜ Su = −

(

)

(

)

⎞

+ S 'u ⎟

⎠

The errors are:

∂(vu ) ∂ (vu ) should be . ∂y ∂x ∂ (uu ) ∂ (wu ) The fourth term on the left hand side of the equation, should be . ∂z ∂x ⎡ ⎤ ∂ ⎡ The second term on the right hand side of the equation, (ν + ν T ) ∂u ⎤⎥ should be ∂ ⎢(ν + ν T ) ∂u ⎥ . ⎢ ∂y ⎣ ∂y ⎦ ∂x ⎦ ∂y ⎣ ∂ ⎡ The third term on the right hand side of the equation, (ν + ν T ) ∂u ⎤⎥ should be ∂ ⎡⎢(ν + ν T ) ∂u ⎤⎥ . ⎢ ∂z ⎣ ∂z ⎣ ∂x ⎦ ∂z ⎦ The third term on the left hand side of the equation,

The correct formulation should read as: ∂w ∂ (uw) ∂ (vw) ∂(ww) ∂ ⎡

∂t

+

∂x

+

∂y

+

∂z

=

(ν + ν T ) ∂w ⎤⎥ + ∂ ⎡⎢(ν + ν T ) ∂w ⎤⎥ + ∂ ⎡⎢(ν + ν T ) ∂w ⎤⎥ + S w = − 1 ∂p + S ' w ⎢ ρ ∂z ∂x ⎣ ∂x ⎦ ∂y ⎣ ∂y ⎦ ∂z ⎣ ∂z ⎦

3.32

Referring to the diagram below, answer the following questions. (a) At the wall, what is the value of u and v? (b) At the symmetry plane, what is the value of u, v and p?

Symmetry Plane u = 1 m/s v=0

p = pressure

Inflow

Outflow

y

Wall x a) At the walls, the value of u and v should be zero velocity because of the no-slip boundary condition. b) At the symmetry planes the exact values may not be known initially. However the condition for a symmetry boundary condition is that the normal gradients of u, v and p and all

other scalar quantities must be zero. ∂p ∂u ∂v e.g. 0 = = = ∂x ∂x ∂x

REVIEW QUESTIONS - CHAPTER 4 4.1

What are the differences between solving a fluid flow problem analytically compared with solving numerically? What are the advantages and disadvantages of each method?

Analytical solutions of fluid flow problems involve closed-form expressions that provide exact solutions of the flow-field variables continuously throughout the domain. This is in contrast to numerical solutions where they provide approximate answers only at discrete points in the geometric domain, for example, at the grid points. The advantages of analytical solutions lie in its exact form, however real fluid flows are generally three-dimensional in nature and analytical relationships are not easily attainable. Even for a simplified three-dimensional flow to a two-dimensional problem, this can be difficult. Numerical solutions are approximations only that are prone to errors in discretization. However, they provide a method to solve real fluid flows that are complex in nature.

4.2

What are the main advantages and disadvantages of discretization of the governing equations through the finite difference method?

Although finite difference can be applied to any type of grid system, the method is more commonly applied to structured grids since it requires a mesh having a high degree of regularity. Finite difference allows the incorporation of higher-order differencing approximations on regular grids which provide a higher degree of accuracy in the solution. However the main disadvantage is that the conservation property is not usually enforced unless special care is taken and the mesh normally requires a high degree of regularity.

4.3 Is finite difference more suited for structured or unstructured mesh geometries? Why? Finite difference is more suited for structured mesh geometries. At each nodal point of the grid, the partial derivatives of the governing equations are replaced by finite difference approximations to yield an algebraic equation for the flow solution at each grid point. The grid spacing between each nodal point needs to be free from high distortion and stretching to maintain accuracy. Finite difference also needs to conform to the constraints of general coordinate systems such as Cartesian grids which can be applied onto structured mesh geometries easily. For unstructured complex shapes, the use of an intermediate coordinate mapping such as the body-fitted coordinate system is needed which complicates the task further.

4.4

Consider the following finite difference formulation in the x-coordinate. φi −1 − 2φi + φi +1 = 0. Δx 2

What is the finite-differencing type? Is it first order or second order? Write the same formulation with respect to the y-coordinate.

The above formulation is a central finite difference for the second order derivative with respect to x evaluated at the point (i). The approximation is second order accurate. The same expression with respect to y is

φ j +1 − 2φ j + φ j −1 Δy 2

4.5

TI = 1

Using finite difference, show that the steady one-dimensional heat conduction equation, T − 2Ti + Ti +1 ∂ 2T k = 0 can be expressed as i −1 = 0? Δx 2 ∂x 2 Ti – 1

Ti

Ti + 1

TI = N

The temperature at point (i + 1) can be expressed in terms of a Taylor series expanded about the point (i) as:

⎛ ∂ 2T ⎞ Δx 2 ⎛ ∂ 3T ⎞ Δx 3 ⎛ ∂T ⎞ Ti +1 = Ti + ⎜ + ⎜⎜ 3 ⎟⎟ +" ⎟ Δx + ⎜⎜ 2 ⎟⎟ 2 x ∂ ⎝ ∂x ⎠ i ⎝ ⎠i ⎝ ∂x ⎠ i 6 The temperature at point (i – 1) can be expressed in terms of a Taylor series expanded about the point (i) as:

⎛ ∂ 2T ⎞ Δx 2 ⎛ ∂ 3T ⎞ Δx 3 ⎛ ∂T ⎞ Ti −1 = Ti − ⎜ − ⎜⎜ 3 ⎟⎟ +" ⎟ Δx + ⎜⎜ 2 ⎟⎟ ⎝ ∂x ⎠ i ⎝ ∂x ⎠ i 2 ⎝ ∂x ⎠ i 6 By summing the above formulations we get

⎛ ∂ 2T ⎞ Ti +1 − 2Ti + Ti −1 ⎜⎜ 2 ⎟⎟ = Δx 2 ⎝ ∂x ⎠ 4.6

What is the second term in the central difference approximation for a first order derivative (given below) called and what does it measure?

⎛ ∂φ ⎞ φi , j +1 − φi , j −1 + O Δx 2 ⎜ ⎟= 2Δx ⎝ ∂x ⎠

( )

The term O(Δxn) signifies the truncation error of the finite difference approximation, which measures the accuracy of the approximation and determines the rate at which the error decreases as the spacing between points is reduced. The above formulation is taken to be second order accurate because the truncation error is of order 2.

4.7

Which of the following, forward difference, backward difference and central difference, is most accurate and why?

Forward and backward difference have a truncation error of order 1, O ( Δx ) . This means that the error decreases as the spacing between points is reduced linearly. A central difference has a truncation error of order 2, O Δx 2 . Therefore it is expected that the central difference will be most accurate in comparison to the forward and backward difference for a given value of Δx.

(

)

4.8

What are the main advantages and disadvantages of discretization of the governing equations through the finite volume method?

The finite volume method discretizes the integral form of the governing equations directly in the physical space. As the finite volume method works with control volumes and not the grid intersection points, it has the capacity to accommodate any type of grid, structured and unstructured which is its main strength. The discretization is a fully conservative method since surface integrals (net inflow-outflow) are applied at the control volume boundaries. One disadvantage of this method compared to the finite difference schemes is that higher-order differencing approximations greater than the second order are more difficult to develop in three dimensions. This is because of the requirement for two levels of approximation, which are interpolation and integration.

4.9 Is finite volume method more suited for structured or unstructured mesh geometries? Why? The finite volume method is suitable for both types of mesh geometries because it uses the integral form of the conservation equations directly in the physical space. For complex geometries where unstructured mesh is used, the mesh can be formed by the combination of triangles or quadrilaterals in the case of two dimensions or tetrahedra and hexahedra in three dimensions.

4.10 What is the significance of the integration of the governing equations over a control volume during the finite volume discretization?

The finite volume discretization involves the integral forms of the governing equations which defines the rate of change of some quantity in a control volume with the net rate of transport across its bounding surface and the rate of production of any sources within that control volume. i.e.

⎛ RATE OF CHANGE ⎞ ⎛ FLUX ⎞ ⎛ SOURCES ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ ⎝ inside control volume ⎠ ⎝ through boundaries ⎠ ⎝ generation - sinks ⎠

fluxW

fluxE source

*flux means the rate of transport through the surface boundary This is the conservation principle which is ensured when applying the finite volume method discretization.

4.11 For the control volume below, Control Volume

δxW

W

δxE

w

P

e

E

Δx show how the one-dimensional steady state diffusion term

∂ ⎛ ∂φ ⎞ ⎜Γ ⎟ is discretised to obtain its ∂x ⎜⎝ ∂x ⎟⎠

⎛ ∂φ ⎞ ⎛ ∂φ ⎞ ⎟ AE − ⎜ Γ ⎟ Aw for central grid nodal point P? ⎝ ∂x ⎠e ⎝ ∂x ⎠ w

discretised equation ⎜ Γ

The control volume discretization of the steady state diffusion term,

∂ ⎛ ∂φ ⎞ ⎜Γ ⎟ is defined as the rate of ∂x ⎜⎝ ∂x ⎟⎠

∂φ flowing across the east and west surface boundaries (flux). Here, the surface areas in the ∂x K normal direction ( n ) to the volume surfaces are resolved with respect to the Cartesian coordinate directions to yield the projected areas Ae and Aw for the east and west faces of the control volume in one change of Γ

dimension. Ae is positive since the normal vectors are projected outwards from the surface area in the same direction as the positive x-axis. Therefore the

⎛ ∂φ ⎞ ⎟ AE ⎝ ∂x ⎠ e

fluxE = ⎜ Γ

and

⎛ ∂φ ⎞ ⎟ AW ⎝ ∂x ⎠ e

fluxW = − ⎜ Γ

fluxW

fluxE P

w

e

This can be shown by applying Gauss’ divergence theorem to the steady state diffusion term. This gives

∂ ⎛ ∂φ ⎞ ∂ ⎛ ∂φ ⎞ 1 ⎜Γ ⎟ = ⎜ Γ ⎟dV ∫ ∂x ⎝ ∂x ⎠ ΔV ΔV ∂x ⎝ ∂x ⎠

=

1 ⎛ ∂φ ⎞ x 1 2 ⎛ ∂φ ⎞ x Γ dA ≈ ⎜ ⎟ ∑ ⎜ Γ ⎟ Ai ΔV i =1 ⎝ ∂x ⎠ i ΔV ∫A ⎝ ∂x ⎠

Here, the projected areas Ai for the one-dimensional case are given by A1 = − AW and A2x = AE . The above equation can thus be written as x

x

⎛ ∂φ ⎞ ⎛ ∂φ ⎞ ⎜ Γ ⎟ AE − ⎜ Γ ⎟ Aw ⎝ ∂x ⎠ w ⎝ ∂x ⎠ e 4.12 In a finite difference scheme, data is resolved at nodal points, how is this different to the finite volume scheme?

The finite difference scheme stores data at the nodal points whereas the finite volume scheme normally stores data at the centre of a control volume as shown below, where the white circles denote the centre of the control volume which stores the data. Interpolation is used to express variable values at the control volume surface in terms of the centre values for fluxes across the boundaries.

δx

W

δx

P

finite difference – data stored at nodal points

E

N n w W

P

e E

Δy

s S y Δx

x

finite volume – data stored at centre of cells 4.13 How is a steady convective-diffusion process different from a pure diffusion process?

A steady convective-diffusion process consists of the fluid in motion and this is reflected through the convective (also called advective) term.

∂ (ρuφ ) − ∂ ⎛⎜ Γ ∂φ ⎞⎟ = 0 ∂x ∂x ⎝ ∂x ⎠

convective

diffusion

In the absence of fluid motion the convective term is zero and the process is pure diffusion. An example of a pure diffusion process is heat conduction through a solid object.

∂ ⎛ ∂φ ⎞ ⎜Γ ⎟ = 0 ∂ x ⎝ ∂x ⎠ diffusion

4.14 Why are upwind schemes important for strongly convective flow?

For convective flows, face values of a control volume, φe and φ w need to specified. Linear interpolation

of node values, leads to central differencing which is second order accurate and exhibits no bias on the

flow direction. Physically the convective process is directional and downstream nodal points are influenced by their corresponding upstream nodes. However the central differencing scheme assigns equal weight to both upstream and downstream nodes. This causes the solution to fluctuate and eventually diverge in strongly convective and hence, directional flow. Upwind schemes recognize the direction of the flow by exerting an unequal weighting influence based on the available variables located at the surrounding grid nodal points to appropriately determine the interface values.

4.15 Why are higher order upwind schemes more favourable than the first order upwind scheme?

The first order upwind scheme is stable and satisfies transportiveness (flow direction), boundness (diagonally dominant matrix coefficients ensuring numerical convergence) and conservativeness (fluxes are represented in a consistent manner). Although this scheme promotes numerical stability, it is widely known to cause unwanted numerical diffusion in space called

false diffusion. In order to reduce these numerical errors, high order approximations such as the second order upwind and third-order QUICK scheme that are widely applied in many CFD problems, have been proposed. The formulation of these schemes is further described in Appendix B. 4.16 For the unsteady convection-diffusion process, what is the difference between explicit and implicit time-marching approaches?

An explicit time-marching approach refers to the difference equation at each time step containing only one unknown and therefore can be solved explicitly for this unknown in a simple manner.

n +1

Time

W

P

E

Properties at level n

n W

P

E

n -1 W

P

Properties at level n+1

E

Properties at level n-1

Illustration of an explicit method. Circles show known values. Squares show the value to be solved.

In an implicit approach, the property φ at the central point P, E and W need to be solved simultaneously. There is more than one unknown value of φ and the unknowns must be obtained through coupling and by means of a simultaneous solution of the difference equations applied at all grid nodal points at a given time level. Implicit methods usually involve the manipulation of large matrices because of the need to solve large systems of algebraic equations.

Properties at level n+1

n +1

Time

W

P

E

n W

P

E

n -1 W

P

E

Properties at level n

Properties at level n-1

Illustration of an implicit method. Circles show known values. Squares show the value to be solved.

4.17 What is the difference between using a direct method and an iterative method to solve the discretized equations?

Direct methods of solving systems of linear equations include Gaussian elimination and the Thomas algorithm. This method solves each unknown sequentially through row operations in a matrix to obtain zeros below the diagonal (an “upper tri-angular matrix”). Iterative methods such as the Jacobi and

Gauss-Seidel method, uses an initial guessed value to solve the set of equations. The resulting values are put back into the set of equations and resolved at the next iteration step. This continues until the solution is converged.

4.18 Is the direct method or iterative method more suitable in solving for a large system of nonlinear equations?

The direct method is only suitable for small systems as it becomes costly when developing an uppertriangular matrix for large systems of equations such as multi-dimensional situations that occur in

practical CFD applications. Iterative methods are more suitable which uses an initial guessed solution, and uses the resulting equations to to systematically improve the solution until it reaches some level of convergence. If the number of iterations is small in achieving convergence, an iterative solver may cost less to use than a direct method. This is usually the case for CFD problems.

4.19 Why does the Gauss-Siedel iterative method converge to a solution quicker than the Jacobi method?

The Jacobi method begins with guessed nodal variables φj at iteration step k = 0 and the nodal

variables φi are treated as unknown at iteration step k +1. The solution for φi, is solved by the iteration step:

φi(k +1) =

n A Bi i −1 Aij (k ) ij − ∑ φ j − ∑ φ (j k ) Aii j =1 Aii j =i +1 Aii

After all the unknowns are solved the first iteration, k = 1, is completed. We proceed to the next iteration step, k = 2, by substituting the iterated values at k = 1 back into our iteration step equation above which yields a new set of values at the next iteration step. This process is continuously repeated for as many iterations as required to converge to the desired solution. The Gauss-Siedel method improves on the Jacobi method by updating nodal variables φ (j k +1) as soon as they are available. In such a case, the previous values of φ (j k ) are replaced by the current values of φ (j k ) , and the iteration step is now given as:

φi(k +1) =

n A Bi i −1 Aij (k +1) ij − ∑ φ j − ∑ φ (j k ) Aii j =1 Aii j =i +1 Aii

Comparing the above two iterative procedures, the Gauss-Siedel iteration is typically twice as fast as the Jacobi iteration.

4.20 What is the technique associated with the successive overrelaxation and why is it used?

Iterative methods can make use of successive overrelaxation to enhance convergence. The basic idea behind this technique is an extrapolation procedure where the intermediate nodal variables φ (j k +1) are further advanced by a given weighted average of the current values of φ (j k +1) with the previous values of φ (j k ) . For example, the extrapolated values for φ (j k +1) is obtained as:

φi (k +1) = (1 − λ )φi(k ) + λφi(k +1) These extrapolated values are continuously used in the system of equations as the iteration process progresses. In the above equation, λ is a relaxation factor whose value is usually found by trial-and-error experimentation for a given problem. Generally, the value of λ is bounded between 0 < λ < 2 in order to ensure convergence. If the successive overrelaxation is used in conjunction with the Gauss-Siedel method, for a value λ between 1 < λ < 2, a significant improvement to the unknowns is obtained at each iteration step is realized and hence convergence is achieved at a faster rate.

4.21 Where are the flow field variables located in collocated grids? How is this different to the locations in a staggered grid?

In a staggered grid arrangement the velocity components are evaluated at the control volume faces while the rest of the variables governing the flow-field, such as the pressure, temperature and turbulent quantities, are stored at the central node of the control volumes. A typical arrangement is shown below on a structured finite volume grid where the arrows depict velocity variables and the white circles depict other stored variables. This arrangement provides a strong coupling between the velocities and pressure, which helps to avoid some types of convergence problems and oscillations in the pressure and velocity fields.

n

N

w W

v e

P

E

S

staggered grid arrangement In collocated grid arrangements, all the flow-field variables including the velocities are stored at the same set of nodal points as shown below. The collocated arrangement offers significant advantages in complicated domains especially the capability of accommodating slope discontinuities or boundary conditions that may be discontinuous.

N

v

u W

P

S

E

collocated grid arrangement 4.22 Write down the formulation of central difference scheme for u velocity in the x direction.

What is its truncation error in terms of ∆x? And state the order of this discretization scheme?

The central difference scheme for the u velocity in the x direction its partial derivative

⎛ ∂φ ⎞ φi +1, j − φi −1, j + O Δx 2 ⎜ ⎟=

2Δx ⎝ ∂x ⎠ Truncation error

∂u is: ∂x

( )

The truncation error is O(Δx2) which signifies second order accuracy. The truncation error measures the accuracy of the approximation and determines the rate at which the error decreases as the spacing between points is reduced.

4.23 What is the purpose of the SIMPLE scheme? Does it give us a direct solution or depend on the iterative concept?

The acronym SIMPLE stands for Semi-Implicit Method for Pressure-Linked Equations. The scheme is used for the calculation of pressure and velocity of an incompressible flow through a guess-and-correct procedure applied to a staggered grid arrangement. When coupled with other governing variables

such as temperature and turbulent quantities, the calculation needs to be performed sequentially since it is an iterative process. The iterative process is shown in the flowchart below.

BEGIN Initial guess p*, u*, v*, φ* STEP 1: Solve the momentum equations (4.63) and (4.64)

u*, v* STEP 2: Solve the pressure correction equation (4.80)

p′

Replace

p* = p, u* = u, v* = v, φ* = φ

STEP 3: Correct pressure and velocity equations (4.65), (4.77) and (4.78)

p, u, v, φ* STEP 4: Solve all other transport equations: Turbulent quantities, Temperature, etc.

φ Check Residuals

No

Solution Converge?

Yes END 4.24 What is the Gaussian elimination method based on? Can this method be used to solve a system of nonlinear algebraic equations?

The Gaussian elimination method is direct method of solving linear systems of algebraic equations.

The algorithm derives from the basis of systematic reduction of large systems of equations to smaller ones. It is a costly method especially for a full matrix containing a large number of unknown variables to be solved. In the case of most CFD problems, that usually result in a large system of non-linear equations, the cost of using this method is generally quite high and direct

methods such as the Gaussian elimination method is not suitable. Instead iterative methods should be used.

4.25 Solve the following set of equations by Gaussian elimination:

200 300 100

-300 200 -400

200 -100 200

T1 T2 T3

=

900 400 600

We apply row operations to convert the first three columns into an upper triangular matrix. This includes the step to prevent ill-conditioning where:

Aii > Aii −1 + Aii +1 This means that the diagonal coefficients are required to be much larger than the sum of the neighboring coefficients.

Swap Row(1) with Row(3), then Row(2) with Row(3)

Row(2) – 2 Row(1) Row(3) – 3Row(1)

Row(3) × ( 145 )

Row(3) – Row(2)

10 20 30

-40 -30 20

20 20 -10

T1 T2 T3

=

600 900 400

10 0 0

-40 50 140

20 -20 -70

T1 T2 T3

=

600 -300 -1400

10 0 0

-40 50 50

20 -20 -25

T1 T2 T3

=

600 -300 -500

10 0 0

-40 50 0

20 -20 5

T1 T2 T3

=

600 -300 200

The matrix is in the form of an upper triangular matrix and back substitution is applied to solve for the unknowns.

10T1 − 40T2 + 20T3 = 600 50 T2 + -20T3 = -300 200 5 T3 = Therefore T3 = 40 1 T2 = {− 300 + 20 × 40} = 10 50 T1

=

1 10

{600 − 20 × 40 + 40 × 10} = 20

4.26 Solve the following set of equations by the Thomas Algorithm:

100 200 100 100

100 -

200 300 100 400

0 300

T1 T2 T3 T4

800 -2000 -200 400

=

In the Thomas algorithm, the first step of the forward elimination process involves eliminating the lower triangular coefficients below the diagonal coefficients to yield zero entries.

Swap Row(2) with Row(3) 800 -200 -2000 400

100 100 200 100

0 100 0 0

200 100 300 400

0 0 0 300

Row(2) – Row(1) Row(3) – 2Row(1) Row(4) – Row(1)

100 0 0 0

0 100 0 0

200 -100 -100 200

0 0 0 300

T1 T2 T3 T4

800 -1000 -3600 -400

Row(4) + 2Row(3)

100 0 0 0

0 100 0 0

200 -100 -100 0

0 0 0 300

T1 T2 T3 T4

800 -1000 -3600 -7600

T1 T2 T3 T4

=

Now we normalize the diagonal elements Ann to unity Row(1) ÷ 100 Row(2) ÷ 100 Row(3) ÷ -100 Row(4) ÷ 300

1 0 0 0

0 1 0 0

2 -1 -1 0

0 0 0 1

T1 T2 T3 T4

8 -10 -36 -76/3

The matrix is in the form of an upper triangular matrix and back substitution is applied to solve for the unknowns.

0 + 2T3 + 0 = 8 T1 + T2 T3 + 0 = -10 0 = -36 -T3 + = -76/3 T4 Therefore T4 = -76/3 T3 = 36 T2 = 26 T1 = -64

4.27 Solve the following set of equations using the Gauss-Seidel method: ⎡12 3 4⎤ ⎡T1 ⎤ ⎡290⎤ ⎢ 1 5 7⎥ ⎢T ⎥ = ⎢105 ⎥ ⎢ ⎥ ⎢ 2⎥ ⎢ ⎥ ⎢⎣ 2 8 2⎥⎦ ⎢⎣T3 ⎥⎦ ⎢⎣130 ⎥⎦

The Gauss-Seidel method requires a matrix with diagonally dominant coefficients. This is where the coefficients on the diagonal must be at least equal to the sum of the other coefficients in that row and at least one row with a diagonal coefficient greater than the sum of the other coefficients in that row.

Rearranging the matrix we get: ⎡12 3 4⎤ ⎡T1 ⎤ ⎡290⎤ ⎢ 2 8 2⎥ ⎢T ⎥ = ⎢130 ⎥ ⎥ ⎥ ⎢ 2⎥ ⎢ ⎢ ⎢⎣ 1 5 7⎥⎦ ⎢⎣T3 ⎦⎥ ⎣⎢105 ⎦⎥

The system of equations is given as: T1 = 112 (290 − 3T2 − 4T3 ) T2 = 18 (130 − 2T1 − 2T3 ) T3 = 1 7 (105 − T1 − 5T2 )

Use the initial guess of T2 = 8 and T3 = 8, the first iteration yields: T1 = 112 (290 − 3(8) − 4(8) ) = 19.5000

T2 = 18 (130 − 2(19.5) − 2(8) ) = 9.3750

T3 = 1 7 (105 − (19.5000) − 5(9.3750) ) = 5.5179 The second iteration yields: T1 = 112 (290 − 3(9.375) − 4(5.5179) ) = 19.9836

T2 = 18 (130 − 2(19.9836) − 2(5.5179) ) = 9.8746 T3 = 1 7 (105 − (19.9836) − 5(9.8746) ) = 5.0919

With further iterations the results converge.

Iteration

T1

T2

T3

1 2 3 4 5 6

19.5000 19.9836 20.0007 20.0003 20.0001 20.0000

9.3750 9.8746 9.9775 9.9959 9.9993 9.9999

5.5179 5.0919 5.0160 5.0029 5.0005 5.0001

Therefore T1 = 20 , T2 = 10 , T3 = 5 4.28 For the same matrix given in Question 4.27, use the Jacobi method to solve the set of equations. Compare the number of iterations for convergence between the Jacobi method and the Gauss-Seidel method.

The system of equations as given from the previous solution is: T1 = 112 (290 − 3T2 − 4T3 )

T2 = 18 (130 − 2T1 − 2T3 ) T3 = 1 7 (105 − T1 − 5T2 )

Use the initial guess of T1 = 10 T2 = 8 and T3 = 8, the first iteration yields:

T1 = 112 (290 − 3(8) − 4(8) ) = 19.5000

T2 = 18 (130 − 2(15) − 2(8) ) = 10.5000 T3 = 1 7 (105 − (15) − 5(8) ) = 7.1429

Using the calculated results, T1 = 19.5000 T2 = 10.5000 and T3 = 7.1429 the second iteration yields: T1 = 112 (290 − 3(10.5000) − 4(7.1429) ) = 19.1067 T2 = 18 (130 − 2(19.5000) − 2(7.1429) ) = 9.5893 T3 = 1 7 (105 − (19.5000) − 5(10.5000) ) = 4.7143

With further iterations the results converge.

Iteration

T1

T2

T3

1 2 3 4 5 6 7 8 9 10 ..

15 ..

20

19.5000 19.1607 20.1979 19.7919 20.1146 19.9264 20.0473 19.9709 20.0187 19.9884 ..

20.0012 ..

19.9999

10.5000 9.5893 10.2813 9.8472 10.1093 9.9366 10.0420 9.9742 10.0165 9.9897 ..

10.0010 ..

9.9999

7.1429 4.7143 5.4133 4.7708 5.1389 4.9056 5.0558 4.9632 5.0226 4.9856 ..

5.0014 ..

4.9999

Therefore T1 = 20 , T2 = 10 , T3 = 5 The Jacobi method takes over 20 iterations to converge compared with 6 iterations using the Gauss-Seidel method.

4.29 Solve the following set of equations using any of the numerical methods discussed. (a)

3x1 −x1 2x1

− + −

x2 2x2 x2

+ + −

3x3 x3 x3

= = =

0 3 2

(b)

10x1 −x1 2x1

− + −

x2 11x2 x2 3x2

+ − − −

2x3 x3 10x3 x3

+ − +

3x4 x4 8x4

(a) x1 = 2 , x 2 = 3 , x3 = −1 (b) x1 = 6197 , x 2 = 202 97 , x3 = 87 97 , x 4 = 117 97

= = = =

6 25 -11 15

4.30 Following the grid arrangement below, derive the following expression:

u − u i −1, j +1 − u i +1, j −1 + u i −1, j −1 ⎛ ∂ 2u ⎞ ⎟⎟ = i +1, j +1 ⎜⎜ 4 Δ xΔ y ⎝ ∂x∂y ⎠ i , j

i-1,j+1

i,j+1

i+1,j+1

i-1,j

i,j

i+1,j

i-1,j-1

i,j-1

i+1,j-1

Δy

∂ ⎛ ∂u ⎞ ∂ ⎛ u i , j +1 − u i , j −1 ⎞ ⎟⎟ ⎜⎜ ⎟⎟ = ⎜⎜ 2Δy ∂x ⎝ ∂y ⎠ i , j ∂x ⎝ ⎠ = = =

∂ ⎛ u i , j +1 ⎞ ∂ ⎛ u i , j −1 ⎞ ⎜ ⎟− ⎜ ⎟ ∂x ⎜⎝ 2Δy ⎟⎠ ∂x ⎜⎝ 2Δy ⎟⎠ u i +1, j +1 − u i −1, j +1

u i +1, j −1 − u i −1, j −1

− 2Δy ⋅ 2Δx 2Δy ⋅ 2Δx u i +1, j +1 − u i −1, j +1 − u i +1, j −1 + u i −1, j −1 4ΔxΔy

Δy

Δx

Δx

REVIEW QUESTIONS 5.1

Why do the results obtained through numerical methods differ from the exact solutions solved analytically? What are some of the causes for this difference?

Analytical solutions are exact solutions based on mathematical principles. However, the governing partial differential equations of fluid flow are complex and cannot be solved by analytical means. The partial differential equations are therefore converted into a system of algebraic equations that are subsequently solved through numerical methods to provide approximate solutions to the governing equations. Some causes for the differences may arise from the discretization methods, round-off errors and mesh resolution.

5.2 In the analysis of CFD results, what does consistency imply?

A formulation is consistency if the system of algebraic equations through the discretization process provides the same (be consistent) solution with the original partial differential equations as the finite quantities, such as the time step Δt and mesh spacing Δx, Δy and Δz, tend to zero. The implication of consistency here is the recovery of the governing equations by reversing the discretisation process through a Taylor series expansion.

5.3 What are the key aspects of consistency?

The discretized equations are an approximation of the real solution due to a truncation of the Taylor series expansion. Essentially, the truncation error represents the difference between the discretized equation and the exact one. As a result, the original partial differential equation is recovered by the addition of a remainder, the truncation error. This error basically measures the accuracy of the approximation and determines the rate at which the error decreases as the time step and/or mesh spacing are reduced. Therefore the key aspects of consistency require the truncation error to become zero when the time step Δt → 0 and/or mesh spacing Δx, Δy and Δz → 0.

5.4

If a system of algebraic equations is equivalent to the partial differential equation as the grid spacing tends to zero, does this also mean the solution of the system of algebraic equations will approach the exact solution of the partial differential equation? Why? In situations when the finite quantities, such as the time step Δt and mesh spacing Δx, Δy and Δz, tend to zero linearly with each other then the solution will be consistent. For example, the ∂u ∂v + = 0 is found to be: discretization of the partial differential equation ∂x ∂y

⎤ u E − uW v N − v S ⎡ + + ⎢O Δx 2 ,Δy 2 ⎥ = 0 2Δx 2Δy ⎢ ⎥ ⎣ Truncation error ⎦ P Here the original partial differential equation is recovered (satisfying consistency) from the discretized equation that includes the truncation error, as the mesh spacing Δx and Δy → 0.

However if the finite quantities in the discretized equation are found linked to each other as follows: ⎡ ⎤ 2 2 ⎡ ∂φ ∂ 2φ ⎛ Δt ⎞ ∂ φ ⎤ ⎢ 2 2 ⎥ − + + , α α O Δt Δx ( )⎥ = 0 ⎢ ⎜ ⎟ 2 ⎥ ∂x 2 ⎝ Δx ⎠ ∂t ⎦⎥ i ⎢ ⎣⎢ ∂t ⎢⎣ Truncation error ⎥⎦ i this may cause problems. Even though the discretized equations might be equivalent to the partial differential equation as the finite quantities shrink to zero, it may not necessarily mean

(

)

that the solution of the discretized equations follows the exact solution of the partial differential equation.

5.5

Explain why the following Taylor series expansion of the one-dimensional 2 2 ⎡ ∂φ ∂ 2φ ⎛ Δt ⎞ ∂ φ ⎤⎥ +α ⎜ = 0 may not show transient diffusion equation: ⎢ − α ⎟ ⎢⎣ ∂t ⎝ Δx ⎠ ∂t 2 ⎥⎦ ∂x 2 i consistent properties. Δt which must → 0 at the The finite quantities Δt and Δx are found together in the form of Δx same rate as Δt, Δx → 0 to achieve consistency. It is also required that Δt « Δx for consistency or else the scheme becomes inaccurate (i.e. if (Δt / Δx)2 is large). For example if the time step size Δt is fixed and we reduce the grid space size of Δx by half, then the solution will not be consistent since (Δt / Δx)2 does not approach zero but instead it gets larger. If however we decrease the time step size Δt then the term will (Δt / Δx)2 approach zero. It is clearly seen that the solution for large (Δt / Δx) is inconsistent with the exact solution when compared with a smaller (Δt / Δx).

5.6 Describe the concept of stability.

The concept of stability involves the growth or decay of errors that are introduced during any stage of the computation. It is noted that the errors being referred here are not those produced by incorrect logic but those that occur because of rounding-off at every step of computation due to the finite number of significant figures the computer hardware can accommodate as well as poor initial guess. A numerical solution method is therefore considered to be stable if it does not magnify the errors that appear in the course of the numerical solution process.

5.7 What is the stability criteria produced by the von Neuman analysis?

The von Neumann analysis is based on predicting whether there will be a growth of error between the true solution of the numerical method and the actual computed solution, which also includes the round-off contamination for linear problems. In general, it is the easiest, most dependable and straightforward method to apply. However it can only be used to establish necessary and sufficient conditions for the stability of linear initial value problems with constant coefficient.

The exact form of the stability criterion depends on the particular differencing approximation applied to the equation. However the process is to analyze the stability of a discretization where the errors introduced at every grid point is given as ξ in = φin − *φ in

φin is the true solution of the numerical method and * φin is the actually computed solution.

For a linear computational algorithm, the error ξ in in the majority of textbooks (Fletcher, 1991, Anderson, 1995) deals with just one term of the finite complex Fourier series, which is given as

ξ in = e at e ik x m

where a is a constant, km is the wave number, t is the time step and x is the grid size.

Formatted: Font: Italic Formatted: Font: Italic

5.8 What is the Courant number and what is its function?

The Courant number is a parameter that relates the velocity with the time step size and the grid spacing size of a discretized formulation which is given as:

C =u

Δt Δx

The Courant number is used in the Courant-Friedrichs-Lewy (CFL) condition where it determines the stability criterion for convection-type equations (i.e. applied to advection schemes). The von Neumann stability analysis produces the CFL condition of: Δt ≤1 C =u Δx This suggests that Δt ≤ Δx / u for the numerical solution to be stable.

5.9

Consider the following discretized equation, locate the Courant number and discuss the Courant-Friedrichs-Lewy condition for stability in this case: φin+1 = φin + α

(

)

Δt φin+1 − 2φin + φin−1 .

2 Δx

The CFL condition in this case is: Δt C = α 2 ≤ 1. Δx

Deleted: Courant number is a function of

C = f (Δt , Δx, α ) .

This means that as long as Δt ≤ (Δx ) is met then the CFL condition will apply.

2

5.10 Provide a definition on the concept of convergence?

If a numerical method can satisfy the two important properties of consistency and stability, we generally find that the numerical procedure is convergent. Convergence of a numerical process can therefore be defined as the solution of the system of algebraic equations approaching the true solution of the partial differential equations having the same initial and boundary conditions as the refined grid system (grid convergence).

5.11

State Lax’s equivalence theorem for convergence. Does it apply to non-linear problems?

For initial value problems governed by the finite difference approximations of linear partial differential equations, Lax’s equivalence theorem states that: “Given an initial valued problem and a finite difference approximation consistency and stability are necessary and sufficient conditions that need to be satisfied for convergence”, i.e. consistency + stability = convergence. However it is must be noted that most computational work for non-linear partial differential equations, as used in CFD, proceeds as though this theorem applies, although it has not been proven directly for this general category of equations.

5.12 What are the three important rules when considering iterative convergence? i. All the discretized equations (momentum, energy, etc.) are deemed to be converged when they reach a user-specified tolerance level at every nodal location. ii. The numerical solution no longer changes with additional iterations. iii. Overall mass, momentum, energy and scalar balances throughout the system are obtained. During the numerical procedure, the imbalances (errors) of the discretized equations are monitored and these defects are commonly referred to as the residuals of the system of algebraic equations, i.e. they measure the extent of imbalances arising from these equations and terminate the numerical process when a user-specified tolerance level is reached. For satisfactory convergence, the residuals should diminish

as the numerical process progresses. In the likelihood that the imbalances grow, as reflected by increasing residual values, the numerical solution is thus classified as being unstable (divergent).

5.13

How is the concept of residual applied to describe the discretized equation of the system of transport equations?

The residuals is a measure of the overall conservation of the flow properties. For any transport variable φ, the discretised form of the partial differential equation can be specifically written as:

a Pφ p = ∑ a nbφ nb + bP where the central coefficient aP should equal to the neighboring coefficients anb which normally depend on the solution of other flow field variables including the time- and spatialvarying fluid flow properties. At the start of the iteration process, the equality in the above equation will not hold and there is a difference. The imbalance variable is called the residual Rp and the above equation can be re-expressed as: RP = ∑ a nbφ nb + bP − a Pφ p 5.14 Differentiate between a local residual and a global residual.

The residual expression at a single nodal point P for one cell volume, which produces a local residual is given as: RP = ∑ a nbφ nb + bP − a Pφ p The global residual R, taken as the sum of each local residual Rp over all the grid nodal points, is monitored:

R=

∑R

P

grid po int s

5.15 What is implied when the residuals become negligible with increasing iterations?

When the residuals reduce with increasing iterations, the simulation is converging.

Convergence is deemed to be achieved when the global residual R satisfies a specified tolerance, i.e. R ≤ ε or ∑ RP ≤ ε . grid po int s

5.16 What is the usual recommended residual tolerance level?

Small tolerance values will incur a large number of iteration steps in reaching convergence.

On the other hand, large tolerance values constitute an early termination of the iteration process for which the numerical solution of the algebraic equations is considered to be rather coarse or not sufficiently converged. Generally, a decrease of the residual by three orders of magnitude during the iteration process indicates at least qualitative convergence. Here, the major flow features are considered to be sufficiently established. Nevertheless, stricter convergence consideration is required for transport variables like energy and scalar species. It is recommended that the scaled energy residual decreases to a recommended convergence tolerance of 10-6 while the scaled scalar species may need to only decrease to a convergence tolerance of 10-5 to achieve energy and species balance respectively.

5.17

Define under-relaxation factor. State its advantages and disadvantages when using a small value.

The under-relaxation is often introduced to stabilize the numerical calculations of the governing equations that are generally non-linear and where the equation of one transport variable is dependent on the others, for example, temperature affecting the velocities in buoyancy flows. The under-relaxation factor α controls the advancement of the transport variable φP during the iteration process. i.e. φ Pnew = φ Pold + α φ Pnew* − φ Pold

(

)

Deleted: u

Formatted: Lowered by 5 pt

A smaller under-relaxation factor will assist in stability and convergence of the iterative process; however it takes a longer time due to the smaller changes at each iteration step.

5.18 Is a converged solution also an accurate solution? Why? A converged solution is not necessarily an accurate solution since accuracy is dependent on the computational setup that can be affected by errors and uncertainties in the numerical calculations. These errors and uncertainties can be generated in either the conceptual modeling or during the computational design phase such as the truncation error occurring during the discretization step.

5.19 Discuss some types of errors that can cause a solution to be inaccurate.

Some of the more common errors that can occur are: o Discretisation error These errors are due to the truncation errors introduced during the discretisation procedure. o Round-off error These errors exist due to the difference between the digital storage accuracy of a computer and the true value of a variable. A single precision solution uses 7 significant figures while a double precision solution uses 15 significant figures. o Iteration or convergence error These errors occur due to the difference between a fully converged solution of a finite number of grid points and a solution that has not fully achieved convergence. o Physical modeling error These errors are those due to uncertainty in the formulation of the mathematical models and deliberate simplifications of the models.

5.20

What are discretization errors? What is the difference between a global error and a local error?

Discretization errors are due to the difference between the exact solution of the modeled equations and a numerical solution with a limited time and space resolution. They arise because an exact solution to the equation being solved is not obtained but numerically approximated. There are two types of discretisation errors: local and global. The local error is the formulation associated with a single step. For this error, the accuracy of the numerical solution concerns mainly on the approximation of the spatial derivative. The solution accuracy for a transient problem however, focuses on the advancement of the transport variable φ through time usually characterized by the global error.

5.21 Which methods can be used to minimize discretization errors?

Discretization errors occur through the transformation of the partial differential equations into algebraic equations where the full Taylor series expansion is truncated. One method to minimize discretization errors is use a higher order discretization scheme such as central differencing (second order) instead of the first order schemes of forward and backward

Deleted: difference between the exact solution of the modeled equations and a numerical solution with a limited time and space resolution.

Deleted: machine

difference schemes. Another method is to reduce the grid or the time step size in order to reduce the truncation error.

5.22

What are round-off errors and what kind of calculations are most affected by them?

Round-off errors occur since there is a limit to a computer’s internal capacity to store digits.

This error is naturally random and in general it is difficult to predict. It depends on the number of calculations, rounding off method, rounding-off type, and even sequence of calculations. Typically however, round-off errors will occur when dealing with decimal values that are limited by the precision. If computer round-off errors are suspected of being significant, one test that can be performed is to employ double precision or on a computer known to store floating point numbers at a higher precision.

5.23 Which method can be used to minimize round-off errors?

Round-off errors are due to the single precision setting during the numerical calculations.

Usually further refinement of the mesh may lead to an increase of the round-off error. One possible way of significantly to reduce this round-off error is to extend the number of significant figures during the iterative procedure to double precision. However the double precision setting is more computationally demanding and hence a longer time is taken per calculation. In practice, some trade-off of the solution accuracy is used to achieve a quicker turnaround of the numerical solution.

5.24 What does it mean to perform a grid convergence (independency) test?

To determine that the discretization error is at an acceptable level a grid convergence test may be performed. This involves decreasing the mesh sizes which leads to an increase in the resolution of the total mesh. The computational results from the initial mesh are compared with the results for the new refined mesh. If the results obtained do not differ significantly, we can conclude that the discretization error is at an acceptable level. But if the values for the transport variables are quite different for this second calculation, then the solution is a function of the number of grid points. In all practical cases, the grid needs to be refined further and the results compared again until a solution is achieved where no significant changes in the results occur. This indicates that the discretization error is reduced to an acceptable error and grid independence is reached.

5.25

What is the difference between verification and validation? Why are these two steps important in analyzing results?

The verification primarily involves the input parameters used for geometry, initial conditions and boundary conditions. This verification step assesses the numerical simulation uncertainty and when conditions permit, estimating the sign and magnitude of the numerical simulation error and the uncertainty in that estimated error.

The validation procedure involves validating the calculations by establishing a range of physical conditions obtained from the calculations, and performing comparisons of the results from the CFD code with experiments that span the range of conditions. These two procedures represent the final phase of the CFD simulation to ensure quality of the results, the credibility the numerical models applied and that the computational model shows an accurate representation of the real physical flow problem.

5.26

Discuss briefly how multigrid methods are employed to increase the computational efficiency of solving CFD problems.

For complex problems that require a large amount of grid points, the time taken to solve the problem is usually a very long time. Multigrid methods can improve the solution time by using a heirachy of where in the simplest case, the coarse ones are just subsets of the fine ones. The main idea behind multigrid methods is the transferring of variables from one grid to another. Usually the variables from a fine grid are transferred to a coarse grid which are solved in quicker time and then the results are slowly refined going from a coarse grid back to the original fine grid. More details on multigrid methods are discussed in Section 8.2.4.

5.27

Discuss briefly how parallel computing is used to achieve computational efficiency.

Parallel computing makes use of multiple computer processors to increase the computational time when compared with single processors. The idea behind parallel computing involves the subdivision of the solution domain into subdomains where each subdomain is assigned to one processor. Since each processor needs data that resides in other subdomains, exchange of data between processors and storage overlap is necessary and specialized algorithms are required.

5.28

Consider the following algebraic equation: apΦp = ∑anbΦnb + b as described in Chapter 4. In matrix form, ap represents the diagonal element while anb is the neighboring element. The condition for convergence stipulates that ∑ a nb a P ≤ 1, which simply means that the sum of the neighboring elements divided by the diagonal elements must be less than unity, at all grid nodal points. Analyze the condition for convergence given the system of equations below.

Case 1:

Φ1 = 0.5Φ2 + 1.5 Φ2 = Φ1 + 2

(1) (2)

by re-arranging the above equations, we have Case 2:

Φ1 = Φ2 =

Φ2 – 2Φ1 –

2 3

(2) (1)

The convergence criteria is: ∑ a nb ≤ 1 ap Case 1: φ 1 = 0.5φ 2 + 1.5 φ 2 = φ1 + 2

a p φ p = ∑ a nb φ nb + b

Therefore a p = 1 and a nb = 0.5 for equation (1) which satisfies the stability criterion.

1 ≤1 5 Similarly a p = 1 and a nb = 1 for equation (2) which also satisfies the stability criterion.

1 ≤1 1

Deleted: ¶

This setup will produce a converged solution. Using the Gauss-Seidel method and setting the variables arbitrarily to 1 we have the following iterations.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

φ1

φ2

1 2 3.5 4.25 4.625 4.8125 4.90625 4.953125 4.976563 4.988281 4.994141 4.99707 4.998535 4.999268 4.999634

1 4 5.5 6.25 6.625 6.8125 6.90625 6.953125 6.976563 6.988281 6.994141 6.99707 6.998535 6.999268 6.999634

φ 1 = 0.5φ 2 + 1.5

φ 2 = φ1 + 2

2 3.5 4.25 4.625 4.8125 4.90625 4.953125 4.976563 4.988281 4.994141 4.99707 4.998535 4.999268 4.999634 4.999817

4 5.5 6.25 6.625 6.8125 6.90625 6.953125 6.976563 6.988281 6.994141 6.99707 6.998535 6.999268 6.999634 6.999817

Case 2: φ1 = φ 2 − 1 φ 2 = 2φ 1 − 3

a p φ p = ∑ a nb φ nb + b

Therefore a p = 1 and a nb = 1 for equation (1) which satisfies the stability criterion.

1 ≤1 1 However a p = 1 and a nb = 2 for equation (2) which does not satisfy the stability criterion because 2 =2 1

This setup will produce a diverging solution. Using the Gauss-Seidel method and setting the variables arbitrarily to 1 we have the following iterations.

φ1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

φ2 1 0 -4 -12 -28 -60 -124 -252 -508 -1020 -2044 -4092 -8188 -16380 -32764

φ 1 = φ 2 − 1 φ 2 = 2φ 1 − 3 1 -3 -11 -27 -59 -123 -251 -507 -1019 -2043 -4091 -8187 -16379 -32763 -65531

0 -4 -12 -28 -60 -124 -252 -508 -1020 -2044 -4092 -8188 -16380 -32764 -65532

-3 -11 -27 -59 -123 -251 -507 -1019 -2043 -4091 -8187 -16379 -32763 -65531 -131067

REVIEW QUESTIONS 6.1 What are some of the benefits of a well designed grid?

A well designed mesh is critical to the quality of the results attained. It removes not only problems that can lead to an apparent instability or lack of convergence but also increases the likelihood of attaining the eventual solution of a CFD problem. It is therefore generally expected that the discretised domain (computational mesh) needs to adequately resolve the important physics and to capture all the geometrical details of the domain within the flow region 6.2 What are some of the advantages of a structured mesh?

Some of the advantages of a structured mesh is that it is easier to create. The points of an elemental cell can be easily addressed by a double of indices (i,j) in two dimensions or a triple of indices (i,j,k) in three dimensions. The connectivity is straightforward because cells adjacent to a given elemental face are identified by the indices and the cell edges form continuous mesh lines that begin and end on opposite elemental faces. It also allows easy data management and connectivity occurs in a regular fashion, which makes programming easy.

6.3

Why is it difficult to write CFD programs that involve a structured mesh for complex geometries?

When the geometry is complex a quality structured mesh is difficult to create due to an increase in grid non-orthogonality or skewness that occurs to accommodate complex regions.

This skewness can cause unphysical solutions due to the transformation of the governing equations. The transformed equations that accommodate the non-orthogonality acts as the link between the structured coordinate system (such as Cartesian coordinates) and the body-fitted coordinate system, contain additional terms thereby augmenting the cost of numerical calculations and difficulties in programming. Because of this, such a mesh may also affect the accuracy and efficiency of the numerical algorithm that is being applied.

6.4 Discuss some of the advantages of an unstructured grid.

An unstructured grid allows cells to be assembled freely within the computational domain.

They handle arbitrary shape geometries especially where domains having high curvature boundaries much better than structured grids which tend to produce highly skewed cells that are undesirable.

6.5

What are some of the difficulties that arise regarding programming of CFD problems for an unstructured mesh?

One difficulty in programming an unstructured mesh is connecting the data points of neighbouring cells. An elemental cell may have an arbitrary number of neighboring cells attached and therefore the programming has to account for this variation. Complex solution algorithms are usually needed to accommodate this which may result in increased computational times. Additionally at wall boundaries the choice of tetrahedaral cells are usually ineffective in resolving wall boundary layers and alternative cell types are needed.

6.6

What conditions and constraints apply if you had to use a structured mesh for the geometry below? What about for an unstructured mesh? Discuss the advantages and disadvantages for this case.

WALL SURFACE

INLET

OUTLET

WALL SURFACE STRUCTURED MESH Vertex

Vertex

Vertex

Vertex Skewed cell at vertex

If a pure structured mesh was used on the geometry shown, high skewness would be observed on vertex points of the circular regions as shown below.

Instead the special case of a block structured mesh in the form of an O-grid mesh should be used where two types of structured mesh blocks are combined. This setting eliminates the skewness that occurs in a pure structured mesh.

UNSTRUCTURED MESH For an unstructured mesh the cells may be assembled freely within the computational domain.

This removes the skewness problem which is found in a structured mesh where the same number of element cells must connect to each other. The downside of this is the programming which has to account for an elemental cell having any arbitrary number of neighboring cells attached to it.

6.7

For the geometry below discuss how using a block-structured mesh has advantages over a single structured or unstructured mesh.

A multi-block structured grid for the above geometry is suitable since it combines a number of mesh blocks to create a mesh that is suited to the geometry. It provides greater flexibility over single structured grids by choosing the best grid topology for each of the sub-divided blocks.

For the above example a block structured grid may be applied as follows:

Additionally the multi-block structured grid also provides the advantage of simpler programming requirements due to the simpler data connection points in comparison with an unstructured mesh. However one disadvantage is the interpolation of results between each structured block during the solving process that is required. This process adds extra computational time to the solution process.

6.8

Why is it more favorable to start off with a coarse mesh when solving a CFD problem?

A coarse mesh can be used to provide quick test runs to evaluate the computer program’s storage and running time. Also it allows the computational setup to be assessed for convergence or divergence of the solution. If the solution is converging then mesh refinement may be undertaken where local refinements can be identified depending on the results. If divergence occurs test runs provides a means of rectifying possible sources of solution errors such as physical modeling and human errors. A fine mesh should not be used for test runs as they often take hours or even days to realize any problems that may occur.

6.9

What is the aspect ratio of a mesh element? Why should a large aspect ratio be avoided, especially in important regions within the flow field? (see diagram below)

Δy

θ

Δx

The "aspect ratio" is a measure of the stretching of a cell, and can be defined from the above diagram as Δy/Δx. Large aspect ratios should always be avoided in important regions inside the interior flow domain as they can degrade the solution accuracy and may result in possible poor iterative convergence (or divergence) depending on the flow solver during the numerical computations.

6.10

What is the skewness of a mesh element? Why is it best to avoid highly-skewed elements?

The skewness of a mesh element describes the amount of distortion and can be measured by the angle between the x and y grid lines. Ideally gridlines should be orthogonal to each other (the angle between the lines θ is approximately 90 degrees). It’s best to avoid highly-skewed elements (where θ < 45 degrees or θ > 135 degrees) as this leads to deterioration of the computational results or numerical instabilities.

6.11

Discuss why tetrahedral elements are a poor choice for meshing near walls and boundaries.

In the near wall and boundary regions tetrahedral elements are a poor choice as it does not resolve the wall boundary layer well. In most cases, the grid yields very long thin tetrahedral cells adjacent to the wall boundaries thereby creating major problems in the approximation of the diffusive fluxes. Instead quadrilateral or hexahedral elements are more effective in this region.

6.12

What techniques can be used in the solver when highly-skewed and high aspect ratio elements exist within the mesh?

When the mesh elements exhibit highly-skewed or have high aspect ratios, the convergence behavior may be hampered due to significant influence of additional terms in the discretised equations. Under relaxation factors can be applied which curtails the iterative advancement of the numerical calculations thereby assists in the convergence of the solution 6.13

A computational domain with different boundary types for the flow around a hydrofoil is shown below. Show where a fine mesh should be appropriately located.

FREE STREAM Hydrofoil INLET

OUTLET

FREE STREAM

A fine mesh is required around the hydrofoil as this is where the fluid flow will change dramatically. The regions far away from the hydrofoil can be coarse and should be far away enough so that it does not affect or have any effect on the main flow region in the centre. An example of the fine mesh near the hydrofoil region is shown below:

6.14

The stretched grid technique is an example of a local refinement technique. What is one problem associated by the application of this technique?

The stretched grid technique is commonly applied to wall boundaries to resolve boundary layers. However care must be taken to ensure the mesh spacing is continuous and that no grid size discontinuities should be minimized as much as possible in regions of large flow changes.

Discontinuity in the grid size destabilizes the numerical procedure due to the accumulation of truncation errors in the critical flow regions.

6.15 Describe how solution adaptation uses an adaptive grid for mesh refinement.

Solution adaptation utilizes an adaptive grid that is intimately linked to the flow field solution.

As the name suggests, the grid is adapted as the flow field develops. It therefore employs the solution of the flow properties to locate the grid nodal points in the physical flow domain.

During the course of the solution, the grid nodal points in the physical flow domain migrate in such a manner as to adapt to the evolution of the large flow gradients. Hence, the actual grid nodal points are constantly in motion during the solution of the flow field and become stationary when the flow solution approaches some quasi steady state condition.

6.16 What are some problems/difficulties in setting up correct boundary conditions?

Some difficulties in setting up correct boundary conditions include: • specification of suitable inlet turbulence intensity, length scales, turbulence kinetic energy and dissipation for a turbulent flow entering the flow domain, • the correct description of the boundary layer velocity profile on the walls and at the inlet, and • the precise distribution of scalar or species concentrations at the inflow boundary.

6.17

What types of conditions may be applied for an inlet boundary and why do you need a corresponding outlet condition?

Specifications for the inlet boundary condition may be in the form of the following: • velocity inlet • mass flow rate • pressure inlet Since an inlet condition is a source of mass entering the system there needs to be a outlet boundary imposed within the flow domain to characterize the mass exiting the system.

Without an outlet boundary, the mass is not conserved which will lead to the CFD calculations in this circumstance to “blow up” rather swiftly.

What is the Neumann boundary condition? Explain how it is used as an outlet boundary condition.

A Neumann boundary condition specifies the values or conditions imposed on the normal derivative at the boundary of a domain such as: ∂φ =α ∂n For an outlet condition based on the Neumann boundary condition, this is given as ∂φ =0 ∂n which means that the convective derivative of the transported variables φ normal to the boundary face is set to zero. The transported quantities at the boundaries are extrapolated along the stream-wise direction of the fluid flow.

6.18

6.19

The geometry for an air-conditioning problem in two rooms separated by a partitioned wall is shown below. Label the boundaries which have to be defined and discuss what types of conditions may be applied.

The boundaries of the model and the partitioned wall should be set to a wall boundary condition. The inlet can be set to a velocity, mass flow rate or pressure inlet. Since the physical problem involves an air-conditioning system the inlet velocity profile may need to be investigated through a sensitivity study of different profiles. An alternative may involve extending the inlet backwards to allow the air to develop naturally therefore leading to a more realistic simulation. The outlet also needs special care as the flow exiting the outlet may not ∂φ observe the Neumann outflow condition of = 0 . In this case the outflow boundary ∂n condition may be allowed to re-enter the domain if an appropriate Dirichlet boundary condition is imposed on the transported quantities

6.20

What requirements must be satisfied for a symmetry boundary condition, and what are the benefits in using this condition?

A symmetry condition is applied to a geometry that possesses some symmetry or periodic property that allows the flow problem to be simplified by solving only a fraction of the domain. This allows a reduction in the number of computational cells used and also the computational effort. At the symmetry plane, the following requirements must be satisfied: • the normal velocity is zero • the normal gradients for all transported properties are zero.

6.21

Discuss the main difference between a symmetry boundary condition and a periodic boundary condition.

A periodic boundary condition is similar to a symmetry boundary condition in that they reduce the computational effort by reducing the entire domain into a smaller repetitive domain.

However the main difference between them is that periodic planes come in pairs with identical mesh distribution in the respective planes. This requirement is necessary since the flow leaving one of the periodic planes is equal to the flow entering in the other paired periodic plane. An example is given below for a mixing tank.

Wall Periodic Planes

Boundary 2

Boundary 1

Flow Direction

Mixing Tank 6.22

The geometry of staggered tube-bank heat exchanger in two-dimensions is shown below. Show by a sketch, how you would define a computational domain for this geometry, by making use of the periodicity or the symmetry boundary condition.

OUTLET

INLET

Symmetry Condition If a symmetry condition is used then a small representative section of the tube-bank that is symmetrical about the edges can be used. The circular sections represent the tubes and therefore would be set as walls. symmetry

OUTLET

INLET

OUTLET

symmetry

Periodic Condition For a periodic condition there needs to be a pair of periodic planes where the flow leaving one of the periodic planes is equal to the flow entering in the other paired periodic plane.

Periodic Plane

Periodic Plane

Flow Direction

6.23

In general, describe how LES (Large Eddy Simulation) solves a turbulent flow.

How does this differ from DNS (Direct Number Simulation)?

Turbulent flow can be characterized by eddies that are present in the flow field. LES solves the turbulent flow by filtering the Navier-Stokes equations to separate large eddies from small eddies. The large eddies are solved directly while the small eddies are approximated. This differs from DNS which solves the entire flow directly without any approximations.

6.24

Why do engineers prefer the Reynolds-Averaged based turbulence models such as the k-ε model over the complex LES model?

Reynolds-Averaged based turbulence models are less computationally expensive which allows quicker turn around times for results. It is less accurate in comparison with LES models however its robustness and ability to handle a variety of flows makes it ideal for applicationsbased research and design that engineers tend to work on.

6.25 What are some of the major problems experienced with the k-ε model?

A major limitation of the k-ε model is the application of the eddy viscosity concept (Bousinessq assumption) which assumes that the turbulent stresses are the same in all directions and hence referred to as an isotropic turbulence model. The model limitations have been identified for flows with separation or swirling behaviour.

6.26

The standard k-ε turbulence model over-predicts the turbulent kinetic energy k when compared with the realizable and RNG turbulence models in the recirculation zones. For the backward facing step case given below, provide an explanation on your final predictions of the reattachment length LR for these turbulence models in comparison with the experimental result (LE is the actual length). In modeling turbulent flow, we effectively solve the following equations: u

∂φ ∂x

+v

∂φ ∂y

⎡ ∂ 2φ ∂ 2φ ⎤ + + Sφ 2 ∂y 2 ⎥ ∂ x ⎣ ⎦

= μt ⎢

a) Use the relationship between μt = ρ C μ k 2 ε

and k to discuss the flow

prediction. b) Which area within the domain requires fine mesh and where can coarse mesh be used? c) What kinds of boundary conditions do we impose for the inlet, outlet and at the wall? d) How should the flow pattern at the outlet be described? e) What are the values of the horizontal u velocity (positive or negative) before, after and at the re-attachment point? f) Sketch to show the development of the velocity profile throughout the flow from the inlet to the outlet at X1, X2 and X3.

g) Keeping the inlet velocity the same, what would be the LE if the working fluid is changed from air to water?

X2 X1 X3

y

INLET ≈

OUTLET x

Step

(Fully Developed)

LE a) The k-ε turbulence model over-predicts the turbulent kinetic energy k which leads to an over-prediction of the local turbulent viscosity, μ t in comparison with the other turbulence models. The reattachment length produced will be shorter than the actual length as the velocity at the walls is slowed down more by the greater shear stress. For the RNG and realizable turbulence model, the turbulent kinetic energy k is underpredicted which produces a smaller μ t and therefore a longer reattachment length is produced.

b) A fine mesh is required in the region where the flow of interest is being analysed and also wall boundaries that have an effect on the bulk flow. A coarse mesh may be applied in regions where the flow is not of interest and has little effect on the main area of analysis.

For the backward facing step example the following setup may be applied.

c) The boundary conditions at the inlet and outlet may be a velocity, mass flow rate or pressure boundary condition. At the wall a no-slip boundary condition is applied.

d) The flow pattern outlet needs to be a fully developed flow. This means that the positioning of the outlet needs to be far away enough from the main flow region to allow the flow to reach its developed state. e) The horizontal component of velocity, u before the reattachment length is negative.

However after the reattachment length it is positive.

f) g) If the fluid is changed from air to water the viscosity increases due to the physical properties of the fluid. A higher viscosity slows down the fluid which allows the separated flow to reattach earlier. Therefore the new reattachment length will be shorter.

6.27 What is y+ and why is it important in context of near wall turbulent modeling?

The y+ value is the non-dimensional distance from the surface of the wall and is defined as: yρuτ y+ =

μ

where uτ = τ w

ρ It is important for turbulent near wall modeling as it determines the flow characteristic within the different regions in a turbulent boundary layer.

What should a typically recommended y+ value for the first adjacent node next to the wall in order to account for the viscous sub-layer?

When applying wall treatments to account for the viscous sublayer in a turbulent boundary layer at the walls, the y+ value should be preferably set to one (i.e. y+ = 1). However, a higher y+ is acceptable so long as it is still well within the viscous sub-layer (y+ = 4 or 5).

6.28

6.29

Without experimental data for turbulent inlet profiles, what is the recommended method to consider turbulence effects?

In many practical CFD problems, specification of the turbulence quantities at the inlet can be difficult and therefore experimental data is not available. For the specification of the turbulent kinetic energy k, appropriate values can be specified through a turbulence intensity I that is defined by the ratio of the fluctuating component of the velocity to the mean velocity. In general, the inlet turbulence is a function of the upstream flow conditions. Approximate values for k can be determined according to the following relationship: 3 (U inlet I )2 2 Similarly, the specification of the dissipation ε can be approximated by the following assumed form: k inlet =

ε inlet = C μ3 4

k3 2 L

If the k-ω model is employed, ω can be approximated by:

ω inlet = 6.30

k1 2 C μ1 4 L

a) For the modeling of flow around a simplified car model given below, state and justify the values of y+ to be used around the car, if “enhanced wall function” model is employed.

Flow around a car may be considered turbulent given the speeds at which cars normally travel at (i.e. 60km/hr). The near wall mesh needs to have a suitable y+ value for the wall function which relates the flow variables to the first computational mesh point thereby removing the requirement to resolve the structure in between, the lower limit of y+ at this point. In the case for turbulent flow over a car, the meshing should be arranged so that the values of y+ at all the wall-adjacent integration points are considered only slightly above the recommended limit, typically between 20 and 30 so that it does not fall into the viscous sub-layer. This procedure offers the best opportunity to resolve the turbulent portion of the boundary layer. b) If Cd, (coefficient of drag) is used for testing mesh independency, on what basis would you think of achieving it? Substantiate your answer with an X-Y plot?

FLOW

WALL SURFACE

XE

A mesh independence test refers to a grid that is refined by increasing the number of grid points until a solution is achieved where no significant changes in the results occur. This indicates that the discretisation error is reduced to an acceptable error and grid independence is reached. If a Cd is used for testing we would check to see where the solution for the value of Cd does not change when the number of grid points has increased. For example, an X-Y plot may show that mesh independence is reached for 1million cells as follows:

0.92

Cd

Drag Coefficient

0.9 0.88 0.86 0.84 0.82 0.8 0

500,000

1,000,000

Grid Points

1,500,000

2,000,000

REVIEW QUESTIONS 7.1

Why do engineers prefer to initially use the standard k-ε turbulence model as a starting point in solving their design problems, instead of using more complex models such as the LES approach? When would an LES approach be used?

The standard k-ε turbulence model provides a solution in a shorter turn around time. This is ideal as a starting point for the design engineer to evaluate the proposed their initial design.

After modifications and for higher quality results a final simulation using the LES approach may be used. In this way less time is lost during the initial testing. The LES model provides a more accurate solution however at a much higher computational cost.

7.2

For the internal pipe flow geometry shown below, formulate appropriate answers to the following questions: d = 0.2 m L1

U = 0.1m/s

L2 d = 0.2 m

a) Define and state all the boundary conditions for this problem.

The boundary conditions for this problem may be defined with a velocity inlet and an outflow condition for the inlet and two outlets respectively. outflow1

velocity inlet

outflow2

b) What would be an approximately suitable length for L1 and L2 to achieve a fully developed flow at the outlets (d = 0.2 m) for this case?

The flow will split into two main flow regions. The flow that continues upwards will have some recirculation and its reattachment length needs to be considered. Therefore length L1 will have to be long enough to account for this reattachment length. A disruption along the axial direction at the intersection will also occur and outflow2 should also be long enough to account for this. In this case L1 and L2 can be set to four times the outflow diameter, D. i.e. L1 = L2 = 4 D = 0.8m

outflow1

L1 inlet

L2 outflow2

c) Show where a fine mesh should be needed.

A fine mesh is required at regions where the flow changes dramatically. In this question this region would be at the intersection where the flow splits into two flow directions. A fine mesh is also required at the walls since the effects of the no-slip condition at the walls will be relevant to the flow field. i.e. outflow1

fine mesh

velocity inlet

outflow2

*there is a fine mesh near the walls using quadrilateral cells which account for the boundary layer at the walls. d) From your understanding of fluid mechanics, sketch roughly what the flow streamlines will look like for this flow problem.

e) Discuss what design ideas may be used to reduce the head loss in this pipe flow and how this can be implemented using CFD. To reduce the head loss due to the fluid separation, the sharp junction where the fluid separates can be eased by applying a fillet to the intersection. This will minimize the head loss created by the flow separation.

smaller separation area fillet

7.3 Explain the Lagrangian description of a fluid motion.

The Lagrangian description of fluid motion observes an individual fluid particle in space as it moves. In this viewpoint, each fluid particle is tracked by its spatial coordinates (x,y,z) and time (t). The fluid properties such as density and velocity are recorded along with the path as time passes. Plotting the position of an individual particle through time gives the pathline of the particle. This can be visualized by sitting in a boat drifting down a river.

The velocity of an individual particle is given as: dx dy dz u= v= w= dt dt dt 7.4

What is the Eulerian description of a fluid motion? How does it differ from the Lagrangian description?

The Eulerian description of fluid motion refers to the entire fluid in the system as a continuum. It focuses on particular points in the space filled by the fluid. This differs from the Lagrangian viewpoint which observes an individual fluid particle in space. For the Eulerian

viewpoint, the fluid properties at each stationary point are given as a function of time. For example the velocity field is governed by: V = V ( x, y, z, t ) 7.5

A streak of dye is released into an internal flow and its motion is tracked and recorded. Is this a Lagrangian or Eulerian measurement?

The motion of the streak of dye has been tracked by allowing the dye to follow the flow path.

This is exactly the same as following an individual particle in the flowing through the fluid and therefore is an example of a Lagrangian measurement.

7.6

A stationary pitot tube is placed in a fluid flow to measure pressure. Is this a Lagrangian or Eulerian measurement?

This is an example of an Eulerian measurement that captures the fluid properties as it varies with time from a fixed point in space.

7.7

The flow in a 90 degree bend is shown below, formulate appropriate answers to the following questions: Bend Entrance d = 0.3

Bend Exit

a) Would a structured or unstructured mesh be better for this geometry? Describe how you would create the mesh?

A block structured mesh would be suitable for this geometry as it allows fine mesh at the walls and a different mesh topology for the curved section. An example is shown below:

b) From your knowledge of the conservation equations of motion, sketch the expected flow streamlines for a flow that has a Reynolds number of 100 and a flow that has a Reynolds number of 10,000.

The streamlines are shown below coloured by velocity magnitude. There is greater colour variation when the flow is at Re=100 due to the viscous effects from the wall producing a laminar velocity profile. The streamlines after the bend are distributed throughout the pipe from the top to the bottom regions. For Re=10,000 the coloured streamlines are more even producing a more evenly distributed velocity profile. After the bend there exists a separation point where recirculation occurs. The streamlines in this region is not distributed throughout the top part of the pipe.

Re = 100

Re = 10,000

c) If a particle with a Stokes number of 0.1 is released in a passive manner from the inlet under laminar conditions, sketch what the expected particle trajectory throughout the pipe would look like? How would the trajectory be different if the flow is turbulent?

Assuming that the Re = 100, a particle exhibiting a small Stokes number will follow the flow streamlines since its particle relaxation time is short enough that the particle can adjust to the changes in the flow field.

St = 0.1, laminar

St = 0.1, turbulent

If the flow was turbulent the same particle flow would occur, in that the particles would follow the flow streamlines with the trajectory dependent on the flow field. e.g. Some disturbed streamlines may be found because of turbulent fluctuations. d) What would the expected particle trajectory be if the particle Stokes number was instead equivalent to 20 for a laminar flow and a turbulent flow?

Particles with excessively high Stokes number exhibit a longer relaxation time which requires greater distances and times for the particle to adjust to the flow field. Therefore when the particle experiences a change in the flow field (i.e. flow changing directions) near the bend the particle’s high momentum prevents the particle from following the streamlines around the bend. Since the particles have such high inertial properties, the influence from the flow field, whether it is laminar or turbulent, would be negligible and hence the particle trajectories would be similar for both flows.

St = 20

e) If you solved this problem in the Eulerian reference, what kind of measurements can be made?

Since the Eulerian technique considers the fluid and particles as separate continuums, the results produced will be based on the fractions of each continuum at a particular point in the flow field. Individual particle tracking is not available in the Eulerian reference.

1.0000 0.9286 0.8573 0.7859 0.7146 0.6432 0.5719 0.5005 0.4291 0.3578 0.2864 0.2151 0.1437 0.0724 0.0010

St = 0.1

7.8

For heat transfer coupled with fluid flow, formulate appropriate answers to the following questions: a) Why is a compact heat exchanger different from a conventional heat exchanger?

Compact heat exchangers are categorically considered by the dimensionless transverse and longitudinal pitches being less than 1.25. Conventional heat exchangers tend to have larger values in comparison.

Longitudinal Pitch

U, Ta D

Transverse Pitch

Computational Domain

Typically, the heat transfer surface areas of compact heat exchangers are substantially increased due to the longitudinal and transverse pitch values which augment the flow, creating various secondary flows and simultaneously destroying the hydrodynamic and thermal boundary layers created by the repetitive nature of the flow. b) Symmetric and cyclic boundary conditions can usually be employed to simplify the flow geometry of a compact heat exchanger. Indicate the boundary conditions for the in-line and staggered arrangements shown below.

In-line arrangement

Staggered arrangement

Symmetry BC

Cyclic BC

Cyclic BC

Symmetry BC

In-line arrangement

Symmetry BC

Cyclic BC

Cyclic BC

Symmetry BC

Staggered arrangement c) The Nusselt number is a ratio of two fluid properties. What are they?

The Nusselt number is a dimensionless number that represents the ratio of the heat transfer rate by convection to the heat transfer rate by conduction. ΔT q conv = hΔT and q cond = k L q conv hΔT hL = = = Nu q cond kΔT / L k d) Explain why higher heat transfer rates are experienced in staggered arrangement over in-line arrangement.

Higher heat transfer rates can be achieved with a staggered arrangement over in-line arrangement of tube banks. This is due to the path of the main flow which experiences greater distortion and extends a greater coverage over the surface areas of downstream tubes thereby resulting in more efficient removal of the heat that are contained within the internal tubes of the compact heat exchanger system.

Streamlines for an inline arrangement

Streamlines for a staggered arrangement

e) What is conjugate heat transfer?

Conjugate heat transfer refers to the fluid flow field and the temperature field within solid bodies. This means that combined fluid-solid heat transfer problems may be solved through different materials. The materials may be of fluid or solid types, all with different properties. d) Why is it important to incorporate the buoyancy effect in natural convection?

What additional modelling effort is required for the transport equations?

Buoyancy must be considered for natural convection as it is the driving force that generates the fluid flow. The flow is induced by the force of gravity acting on the density. No major additional effort is required since the buoyancy force is only dependent on the density differences and hence the specific gravity of the fluid that is caused by the differences in temperature. e) What is radiation heat transfer? How does it affect the surface solid temperature (see section 7.4.1.2)?

Radiation heat transfer occurs through electromagnetic waves emitted from one body to another. The type of radiation that causes heat transfer is called the thermal radiation. The electromagnetic waves are emitted from an energy source to the surface of a solid which absorbs these waves. This energy transfer causes the temperature at the surface to increase.

7.9

The Ahmed model is often used to model bluff car bodies. The schematic for one model is shown along with two cars travelling in line together with the mesh geometry shown below. Analyse this problem by answering the following questions: FLOW

WALL SURFACE

XE

Trailing Lead

a) Explain why there is a fine mesh surrounding the near regions of each car body.

A fine mesh is required to resolve the boundary layers caused by the surface of the car bodies. Additionally flow separation and wake effects need to be resolved in this region. b) Indicate the regions of high and low pressure encountered around the car bodies. low pressure low pressure high pressure

c) What is the general effect that the lead car model has on the drag coefficient of the trailing model?

The flow that separates from the leading edge of the rear slant of the lead model creates a flow impingement on the front portion of the trailing model. The low pressure in the wake region of the lead car will reduce the pressure drag of the rear model.

d) What happens to the drag coefficient if the distance between the two models is in close proximity (i.e. x/L ≈ 0.25 where x is the distance and L is the length of the models)?

When the two models are in close proximity to each other, the drag coefficient of the lead model is considerably lower when compared to that of the trailing model. This effect is caused by the apparent presence of another model in its wake. The mere presence of this trailing model increases the base pressure of the lead model, which subsequently reduces the drag of the lead model. e) While using two different k-ε turbulence models, one model over-predicts turbulent kinetic energy k while the other under-predicts it, explain the difference of the flow behavior when using these two models against the accurate experimental value XE, shown in the figure?

An overprediction of the turbulent kinetic energy k will lead to an overprediction of the effective viscosity since μ T ∝ k 2 . The higher viscosity value will shorten the reattachment length XE due to the viscous effects that retard the flow momentum quickly. On the other hand, an underprediction of k will lead to a longer reattachment length since the effects of a lower viscosity will allow the fluid momentum to flow more freely.

7.10

Two particle trajectory profiles are shown below for mono-sized particles released from the bottom inlet. The particle velocity magnitude has the higher value in Fig. (a) and the lower value in Fig. (b). The initial particle velocity is a factor of ten of the air flow velocity, i.e. U* = 10 = Uparticle / Uair flow. a) What would be the approximate particle Stokes number for the flow path in (a) and for the flow path in (b) shown in the figures below?

The particle trajectories shown in (a) are straight which reflect the particles high momentum and an inability to follow the flow streamlines. An approximate Stokes number for this particle would be St >> 1 . Conversely the particle trajectories shown in (b) begin directed upwards but are able to curve around as the geometry curves. An approximate Stokes number for this particle would be St < 1

a

b

b) Explain why there is a difference between the particle trajectories obtained for flow path (a) and for flow path (b)? (Hint: compare the velocity profiles?)

The difference in particle trajectory can be explained through the velocity profiles. In figure (a) the particle’s velocity when it reaches the top of the geometry remains high.

This suggests that the particle momentum is high and the influence from the flow field is negligible ( St >> 1 ). The particle trajectory for (b) shows that the velocity has decreased which allows the particles to adjust to the flow field. This suggests that the particle momentum is a lot smaller ( St < 1 ) which enables the flow field to influence the particle trajectory.

7.11

Formulate appropriate answers to the following questions for high speed flows: a) Why is the Direct Numerical Solutions (DNS) technique very useful for studying supersonic flows?

DNS techniques are very useful for studying supersonic flows as they are able to provide details regarding the onset of flow instability and subsequent transition to turbulence. The full compressible three-dimensional Navier-Stokes equations can be solved to accommodate the spectrum of varying length scales that exist within the complicated fluid phenomenon. b) How does the outflow condition in supersonic flows differ from the usual Neumann boundary condition in subsonic flows?

The Neumann boundary condition in supersonic flows is not influenced by downstream conditions which differs from the usual Neumann boundary condition in subsonic flows. The properties are now calculated based on an extrapolation from the upstream conditions in contrast to imposing the zero normal gradient constant. c) For the following supersonic flow over a flat plate, discuss what types of boundary conditions are suitable to capture the flow features.

Computational domain

Ma = 3 Leading Edge

Trailing Edge

L Inflow / Upper boundary Dirichlet condition based on free stream values

Computational domain

Plate surface No-slip condition for velocity components Dirichlet or adiabatic condition for temperature Leading Edge

Outflow boundary All properties are extrapolated from upstream

Trailing Edge

d) Discuss the effects of using a constant wall temperature instead of an adiabatic wall in terms of the boundary layer development and its effect on the temperature profile in supersonic flows.

An adiabatic wall increases the boundary layer temperature by a substantial margin above the constant wall. This is primarily due to a relatively lower density which leads to a thicker boundary layer and consequently a broader shock layer. Since the temperature gradient is zero at the wall for the adiabatic case, temperatures within the thermal layer are expectedly to be higher than those of the constant wall temperature case. Essentially, a colder wall temperature suppresses the boundary layer flow. e) Discuss the advantages and disadvantages of using DNS methods over theoretical approaches such as the Parabolic Stability Equation (PSE) calculations.

An advantage of DNS over PSE methods is the prediction of the onset of breakdown to turbulence where the nonlinear interaction is beyond the capability of PSE. However, a difficulty is prescribing a suitable inflow boundary condition which is of paramount importance. Therefore PSE calculations are still beneficial in providing solutions across the range of the fluid flow from laminar state to the early and later stages of transition to capture the manifestation of chaotic structures in a fully developed turbulent flow.

f) For the following flow over an aerofoil, discuss why the following grid is necessary.

The grid is an example of a multi-block grid that offers flexibility in generating dense regions around the airfoil while a coarse mesh can be applied away from the airfoil. The dense region is required as it needs to capture the developing boundary layer along the chord line between the leading and trailing edges and expanding shock layers along the aerofoil. g) When the angle of attack of the NACA0012 aerofoil is at 5o, what happens to the pressure over the bottom surface and how does this affect the temperature (see section 7.4.5.2)?

At a 5o angle of attack, the pressure has significantly increased when compared with a 0o angle of attack. A smaller boundary layer thickness is produced and hence the vorticity is small. The pressure gradient is favorable and an upward resultant force (the lift force) occurs. The temperature across the bottom surface is significantly higher than the temperatures over the top wing surface when compared to the case of 0o angle of attack.

REVIEW QUESTIONS 8.1

Simplify the general continuity equation below to a steady incompressible flow equation: ∂ρ ∂ ( ρ u ) ∂ ( ρv ) ∂ ( ρ w ) ∂t

+

∂x

+

∂y

+

∂z

=0

By applying the chain rule, the above equation can be rewritten as: ∂ρ ∂u ∂ρ ∂v ∂ρ ∂w ∂ρ +ρ +u +ρ +v +ρ +w =0 ∂t ∂x ∂x ∂x ∂y ∂x ∂z ∂ρ ∂ρ ∂ρ ∂ρ For incompressible flow the derivative of ρ is zero Æ = = 0, =0 ∂x ∂y ∂z ∂t Therefore the continuity equation becomes: ∂u ∂v ∂w + + =0 ∂x ∂y ∂z 8.2

How is the pressure term used to satisfy the continuity equation in the markerand-cell (MAC) method?

The marker-and-cell method uses the pressure term as a mapping parameter to satisfy the continuity equation. ∂u i =0 ∂xi ∂u i ∂u ∂ 2ui 1 ∂p + ui i = − +ν ∂t ∂xi ρ ∂xi ∂xi2 Taking the divergence of the momentum equation, the Poisson equation for the pressure can be obtained according to: 1 ∂ 2 p ∂h ∂ ∂u i = − ρ ∂xi2 ∂xi ∂t ∂xi where hi = ν

∂ 2ui ∂u − ui i 2 ∂xi ∂xi

8.3 Discuss briefly the idea behind the fractional step procedure.

The fractional step procedure obtains an auxiliary velocity field from solving the momentum equation in which the pressure-gradient term can be entirely excluded or computed from the pressure in the previous time step. The pressure is then computed through a Poisson equation similar to the MAC method that maps the auxiliary velocity onto a divergence-free velocity field.

8.4 What types of applications and situations involve compressible flows?

Compressible flows occur when the fluid density changes are significant throughout the flow.

They are important in:

• •

aerodynamic applications – convergent-divergent nozzles, aerofoils and high speed applications involving shock waves. turbomachinery applications – supercharger screws,

8.5

What difficulties arise from modeling a transient supersonic flow around an airfoil?

High speeds are often encountered over airfoils which result in very high Reynolds number and turbulence effects are mainly concentrated within the thin boundary layers at the airfoil surface. This requires a very fine mesh which near the walls which greatly limits the time step size for computation; this in turn dramatically increases the computational burden to achieve a solution and very small time steps or implicit methods are required.

8.6

What is the biggest difficulty that has to be overcome with compressible flows involving shocks?

For high speed flows, the effects of compressibility cannot be neglected. These flows have very high Reynolds numbers and the turbulence effect is mainly concentrated within the thin boundary layers. The presence shocks in the flow produce oscillations in the iteration process which need to be overcome. Additionally the full compressible Navier-Stokes equations need to be used for compressible flow which is difficult for resolving the thin viscous boundary layers around surfaces in high speed flows.

8.7

What techniques can be used to minimize oscillations that occur in compressible flow due to discontinuities at the shock front?

To avoid the problem of oscillations especially due to the discontinuity at the shock front, the concept of artificial dissipation can be introduced into the equations. Examples of this technique are found in the total variable diminishing (TVD) and the Essentially Non Oscillatory (ENO) schemes which are well suited for capturing shock waves How does a higher order scheme such as a 5th order scheme, deal with discontinuities?

Higher-order schemes such as the fifth-order method MP (Monotonic Preserving) schemes satisfy not only the monotonic variation of velocity u but also the relizability condition near the extrema. It addresses the problem of upwind methods where the narrow stencils cannot distinguish between shocks and extrema. The MP scheme is highly accurate in smooth regions of the flow and has sharp non-oscillatory transitions at discontinuities.

8.8

8.9

Under what circumstances, would adaptive meshing be used? What would happen if a fixed mesh is employed instead?

Adaptive meshing is generally used for mesh refinement in areas where the flow field becomes complex during a simulation. Therefore adaptive meshing is constantly changing and dependent on the flow field. For strong migrating shock fronts, adaptive meshing may be applied to resolve the sharp discontinuities that exist within the flow domain as the solution changes continuously through time. If the mesh remained fixed, then the solution may not be able to account the sudden shock fronts.

8.10 Discuss briefly the concept behind the r-refinement grid adaptive technique.

The concept behind the r-refinement is to physically relocate the original non-adapted mesh to the time varying changes of the solution in the domain, based on some criteria for adaptation whilst maintaining the identity and data structure of the domain being solved.

However, the success of relocating the nodes or re-meshing depends on the criteria used to properly guide the node location.

8.11 What kinds of applications commonly use moving grids?

Some applications that use moving grids include: • an internal combustion engine where deforming grids permit the simulation of the piston and valve motion • stir-tanks where a rotating impeller is simulated • rotor-stator interaction in turbomachinery • melting ice where the ice-water interface is simulated 8.12

Explain how a moving grid can be applied to simulate a screw supercharger shown below. What parts would need to remain stationary and what parts would be allowed to move?

The internal rotors of the supercharger both need to be moving for the airflow to pass through the screw. Therefore both the male and female rotors need to have a moving mesh applied to it. An outer casing (which is not shown) is required to enclose the system and the air region within the casing needs to be a fixed mesh. outer casing (stationary) male rotor (moving)

female rotor (moving)

8.13

What is the main advantage in the numerical solution of using multi-grid methods in terms of the handling of high and low frequency errors?

During a numerical solution, a whole spectrum of errors can propagate throughout a flow field. The advantage of the multigrid method enhances the damping of numerical errors through the flow field. Iterative procedures such as the Jacobi and Gauss-Siedel are implemented within the multigrid which removes high frequency errors in a few iterations.

Removal of low frequency errors on fixed grids causes slow convergence in iterative

methods. However by progressively moving the results to coarser grids, the low frequency errors are essentially damped and when these results are transferred back to the fine grid, the low frequency errors are much smaller than they would have been for an equal number of sweeps that are performed on the fine grid itself which leads to faster convergence.

8.14

Explain what domain decomposition and load balancing are in parallel computing.

Parallel computing involves performing simultaneous operations of multiple computational tasks on many computers. The computational domain needs to be split into partitions for each computer to solve. Domain decomposition involves distributing the domain into separate partitions for each computer. Uniform computational activities on all processors must be maintained and is known as load balancing.

8.15

What is the immersed boundary method and how is this different from using a boundary fitted grid?

The immersed boundary method is an innovative approach to deal with the problem of modeling fluid flows where fixed Cartesian grids can be applied. This is most significant for non-conformal bodies. If a body fitted grid is applied the coordinate system would need to be transformed back to a Cartesian grid through some mapping functions. In the immersed boundary method, a Cartesian grid can be applied without regard to the body’s geometric shape. Therefore, the solid body cuts through this Cartesian volume grid. Because the grid does not conform to the solid boundary, appropriate boundary conditions are incorporated into the system by modifying the governing equations in the vicinity of the boundaries.

8.16

What is a direct numerical simulation (DNS)? How does it differ from a Reynolds Averaged Navier-Stokes (RANS) approach in terms of its handling of turbulence?

DNS solves the Navier-Stokes equations completely without any modeling. The computations resolve all relevant spatial and temporal scales and accommodate the entire range of length scales including the smallest scales of which the viscosity is active. This differs from RANS based simulations which uses models based on the averaged flow field to solve turbulent features.

8.17

What are the Kolmogorov microscales? How do these scales impact on the mesh design?

Kolmogorov microscales are the smallest scales of turbulence that exist in all turbulent flow.

There are three scales that are considered; length, time and velocity scales which are dependent on the fluid properties. These microscales vary with the Reynolds number. As the Reynolds number increases the difference between the largest and smallest length scale increases and the number of grid points required to resolve turbulence increases as Re 9L 4 .

8.18 Why can’t DNS be used to solve high Reynolds numbers flows at the moment?

DNS for high Reynolds numbers are not applicable at the moment because of the extreme demands on computational resources due to the high number of grid points required. Typical Kolmogorov length scales are on the order of 0.1mm and frequencies around 10kHz. For a 1m x 1m 2-D block a grid size of 10,000 2 = 1x10 8 is required. Additionally for an unsteady case, 10,000 time steps are required to resolve 1second of flow.

8.19

What is the main concept behind LES in turbulent modeling?

The main concept behind LES is to solve only for the large eddies explicitly and to apply a model to the smaller and more universal eddies. This concept is based on the premise that the largest eddies interact strongly with the mean flow and contain most of the energy that contribute to the main effects of turbulence. Smaller eddies are more universal and tend to be isotropic and therefore a model can be used to account for this.

8.20

What are the subgrid scale (SGS) models in LES? How are they used to define small scales of turbulence?

A filtering function is applied to the flow field to separate the large and small eddies. For a finite volume method it is natural to define the filter width from the average grid volume. In a rough sense, the flow eddies larger than the filter width are large eddies while eddies smaller than the filter width are small eddies which require modeling. A subgrid scale models such as the Smagorinsky model is required to close the equations since additional terms appear in the subgrid scale.

8.21

In the RANS-LES coupling approach, in which region of the mesh would you apply the RANS model and which region would you apply the LES model?

The RANS-LES coupling approach takes advantage of the well established RANS turbulence models for near-wall predictions instead of a full resolution through LES. However since the RANS approach is inadequate in accounting for the turbulent fluctuations away from the wall, LES is used. Therefore the RANS-LES coupling approach combines both methods where the inner near-wall region is handled by the RANS approach while the outer region of the bulk flow is solved through LES. 8.22

Why would you use a RANS-LES coupling approach to model high Reynolds number turbulent flows?

The RANS-LES approach provides a quicker solving time to a pure LES and DNS approaches. By making use of the RANS based near wall turbulence modeling a smaller mesh size can be used in comparison with LES and DNS in this region.

8.23

What is the difference in one-way coupling and two-way coupling in multiphase flows?

One-way coupling refers to one phase affecting the other phase while there is no reverse effect. For example a gas-particle flow which as a very dilute concentration of particles, will have a one-way coupling where the gas affects the particles only. However if there is a mutual effect between both phases then the flow is two-way coupled.

8.24

Would you use an Eulerian-Eulerian or Eulerian-Lagrangian for a multiphase flow that had a high mass loading for the secondary phase (i.e. not the continuous phase)?

When the mass loading is high, the influence of particles on the fluid motion needs to be properly accounted. A two-way coupling requires the computation of particle trajectories and fluid flow to be simultaneously carried out and therefore an Eulerian-Eulerian approach would be more suited. The Eulerian-Lagrangain approach is more suited for dilute flows with one-way coupling.

8.25

Combustion is a complex phenomenon to model. What types of considerations must be made in modeling combustion?

Combustion modeling is extremely complex. The model needs to consider the chemical reactions that are strongly coupled with turbulence effects which occur within the thin reaction zone of the flame. Some important considerations are: There exists various degrees of mixing and the turbulence has a wide range of time and length scales.

The chemical reactions of combusting species are highly non-linear; they strongly affect the flow variables such as temperature and density which can change rapidly by a factor of eight through the flame region.

A combustion process usually comprises of a large number of species partaking in many reaction steps, which may result in having very different complicated transport properties. The combustion time and length scales are typically smaller than the smallest turbulent scales, making them difficult to resolve by numerical or experimental means.

8.26

Explain fluid-structure interaction (FSI) modeling. In what applications can this be used?

FSI involves the interaction caused by a force balance between a flexible structure and the surrounding fluid. The concept is to synthesize independent computational approaches of the fluid dynamics and the structural dynamic systems through an interface that couples the two models. Some applications of FSI include stability analysis of airplane wings, design of bridges and flow of blood through arteries.

8.27

What is the key requirement (and the most complex) that enables the interaction between the fluid and structure in FSI? For a fully coupled model, the CFD governing equations are reformulated by combining the fluid and structural equations of motion which are solved and integrated in time simultaneously. This is an integral requirement that is difficult to achieve since there are severe limitations because of the need to solve the equations in two different reference systems: fluid equations as Eulerian and structural equations as Lagrangian.

8.28

What advanced techniques would be required to simulate airflow through the respiratory system into the lungs? What about pulsating blood flow through veins and arteries?

Advanced medical imaging techniques such as magnetic resonance imaging (MRI) along with high level geometry and meshing computational software is enabling the construction of anatomically accurate geometric models such as the respiratory systems. The airflow can exist in all three flow regimes, laminar, transitional and turbulent. This means advanced methods to accommodate these flows such as RANS-LES methods are required. For blood flow through veins and arteries, the viscoelastic properties of blood need to be modeled by setting the blood as a non-newtonian fluid. FSI are also required to simulate the forces applied to and from the artery walls.

8.29 Briefly discuss the concept of the Lattice Boltzmann method.

The Lattice Boltzmann method (LBM) is a methodology based on the microscopic particle models and mesoscopic kinetic equations. It is a discrete computational method based upon the Boltzmann equation instead of the Navier-Stokes equations. It considers a typical volume element of fluid to be composed of a collection of particles that are represented by a particle velocity distribution function for each fluid component at each grid point. By simulating the interaction of a limited number of particles the viscous flow behavior emerges automatically

from the intrinsic particle streaming and collision processes. The rules governing the collisions are designed such that the time-average motion of the particles is consistent with the Navier-Stokes equation.

8.30 Briefly discuss the concept of the Monte Carlo method.

The direct simulation Monte Carlo (DSMC) method is used in computational fluid dynamics for solving high Knudsen number (Kn) flows. It is a direct particle simulation method based on kinetic theory. The fundamental idea behind this method is to track a large number of statistically representative particles. The particles’ motion is later used to modify their positions, velocities, or even chemical reactions in reacting flows. Conservation of mass, momentum and energy are enforced on the requirement of the machine accuracy. The primary approximation of the DSMC method is to uncouple the molecular motions and the intermolecular collisions over small time intervals. Here, particle motions are modeled deterministically while the collisions are treated statistically.

8.31 Briefly discuss the concept of the Particle method.

The concept of particle methods for the simulation of the continuum phenomena is obtained through the solution of ordinary differential equations (ODEs) that determine the trajectories and the evolution of properties carried by the particles. The method is based on a Lagrangian set of equations which mean that is not confined to mesh problems. The two popular particle methods, Vortex and SPH both involves the approximation of the Lagrangian form of the Navier-Stokes equations by replacing the derivative operators through equivalent integral operators, which are in turn discretised on the particle locations.