Lecture2_CFD_Course_Governing Equations [Compatibility Mode].pdf

8/14/2019 Lecture2_CFD_Course_Governing Equations [Compatibility Mode].pdf

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2

Introduction

CV, or integral, forms of equations are useful fordetermining overall effects.

However, detailed knowledge about the flow fieldinside the CV cannot be obtain. (motivation for

differential analysis)Application of differential equations of fluid

motion to any and every point in the flow field or

flow domain


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3

Description of fluid motion

FLUID KINEMATICS - the study of how fluids flowand how to describe fluid motion.

There are two distinct ways to describe motion.

Lagrangian description of fluid motion: Individual

objects or individual fluid particles are tracked (function of

time) as they move through the flow field.

Eulerian description of f luid f low:A finite volume called a

flow domain or control volume (CV) is defined, throughwhich fluid flows in and out and a track of the position and

velocity of a mass of fluid particles is not made. Instead,

field variables, are defined as functions of space and time,

within the control volume.


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4

Description of fluid motion


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Lagrangian Description

Named after Italian mathematician Joseph LouisLagrange (1736-1813).

Lagrangian description of fluid flow tracks theposition and velocity of individual particles.

Based upon Newton's laws of motion. Difficult to use for practical flow analysis.

Fluids are composed ofbillions of molecules.

Interaction between molecules hard to describe/model.

However, useful for specialized applications

Sprays, particles, bubble dynamics, rarefied gases.

Coupled Eulerian-Lagrangian methods.


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Eulerian Description

Named after Swiss mathematician Leonhard Euler (1707-1783).

Eulerian description of fluid flow: a flow domain or control

volume is defined by which fluid flows in and out.

Field variables are define which are functions of spaceand time.

Velocity,

Acceleration,

These (and other) field variables define the flow field.

Well suited for formulation of initial boundary-value

problems (PDE's).

, , , , , , , , ,V u x y z t i v x y z t j w x y z t k

, , , , , , , , ,

x y z

a a xyzt i a x yzt j a xyzt k

, , ,a a x y z t

, , ,V V x y z t


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Conservation of Mass (Differential CV & Taylor series)

Infinitesimal CV

of dimensions

dx, dy, dzArea of right

face = dy dz

Mass flow rate through

the right face of the

control volume


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Now, sum up the mass flow rates into and out ofthe 6 faces of the CV

Plug into integral conservation of mass equation.

After substitution,

Net mass flow rate into CV:

Net mass flow rate out of CV:


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Dividing through by volume dxdydz

Or, if we apply the definition of the divergence of a vector

Use product rule on divergence term

orWhere, D/Dt is

material or total

derivative


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Total or Material derivative The total derivative operator (d/dt) is also given special notation,

D/Dt.

Remember D/Dt (or d/dt) and /t are physically and numericallydifferent quantities, The former is the time rate of change

following a moving fluid particle while later is the time rate ofchange at fixed location.

Provides ``transformation'' between Lagrangian and Eulerianframes.

Other names for the material derivative include: total, particle,

Lagrangian, Eulerian, and substantial derivative. It can be applied to any fluid properties, both scalars and vectors

(e.g. V, , p etc.). For example, the material derivative of

velocity & pressure can be written as:


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Conservation of Mass (Cylindrical coordinates)

In general, continuity equation cannot be used by

itself to solve for flow field, however it can be used to:

1. Determine if velocity field is incompressible.

2. Find missing velocity component.


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Conservation of Mass (Special Cases)

Cartesian

Cylindrical

Incompressible flow: = constant

Steady compressible flow:

0

Also D/Dt =0


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Where n is unit normal vector. By applying Gauss

divergence theorem , volume integral of divergence of

vector can be equated to area integral over the surface that

defines the volume,

Continuity Equations - integral

SV

ndSVdVdtd .

SV

ndSVdVVdivdt

d. 0)(

V

dVVt

Any size of volume, V 0)(

Vt

Mass conservationequation

In Cartesian coordinate system, it

expressed as:

0

z

w

y

v

x

u

t


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Momentum Equation

Consider a fluid particle and Newton's second law,

Acceleration Field - The acceleration of the particle is the

time derivative of the particle's velocity.

However, particle velocity at a point is the same as the fluid

velocity,

To take the time derivative of, chain rule must be used.

particle particle particleF m a

particleparticle

dVa

dt

, ,particle particle particle particleV V x t y t z t

particle particle particle

particle

dx dy dzV dt V V V a

t dt x dt y dt z dt


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Acceleration Field

Since

In vector form, the acceleration can be written as

The total or material derivative operator d/dt ( or D/Dt)

emphasize that it is formed by following a fluid particle as it

moves through the flow field.

First term is called the local acceleration and is nonzeroonly for unsteady flows.

Second term is called the advective acceleration and

accounts for the effect of the fluid particle moving to a new

location in the flow, where the velocity is different.

, , , dV V

a x y z t V V dt t

particle

V V V V a u v w

t x y z

, ,particle particle particledx dy dz

u v w

dt dt dt

.


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Momentum Equation

The Newton's second law application to differential fluid

element in a CV is given by:

Total forces on differential fluid element:

While body forces are analyzed by simple relation such as:

Surface forces are not as simple to analyze as above since

they consist of both normal and tangential components.

(1)


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Forces acting on CV consist of body forces that actthroughout the entire body of the CV (such as gravity,

electric, and magnetic forces) and surface forces that

act on the control surface (such as pressure and viscous

forces, and reaction forces at points of contact).

Body forces act on each

volumetric portion dV of

the CV.

Surface forces act on eachportion dA of the CS.

Momentum Equations


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Body Forces (gravity, electric and magnetic forces)

The most common body forceis gravity, which exerts a

downward force on every

differential element of the CV.

Total body force acting on CV:

The differential body force:

Typical convention is that

gravity acts in the negative

(e.g. z-direction):


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Surface Force Surface forces are not simple to analyze

as they include both normal & tangentialcomponents, i.e. the description of the

force in terms of its coordinate changes

with orientation.

Second-order tensor called the stress

tensorij is used in order to adequatelydescribe the surface stresses at a point

in the flow.

Diagonal components xx, yy zz are

called normal stresses and are due to

pressure and viscous stresses. Off-diagonal components xy, xz etc.,

are called shear stresses & are due

solely to viscous stresses.

Total surface force acting on CS:

Surface integrals are

cumbersome to solve


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Momentum Equation

For brevity, here we consider on only x-component of Totalforces acting on differential element to simplify the diagram.

Thus, the body force and the net surface force due to

stresses in the xdirection can be given as:

Combining equations above, equating to Total forces in x-

direction (eq. 1) & dividing by dxdydz becomes:

(2)


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Momentum Equation

Thus the differential forms for y and z- direction momentum

equations :

Or equations 1, 2, 3 & 4 are combine to give equationsbelow:Cauchys

Equation

(3)

(4)


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Momentum Equations (integral)

Body Force Surface Force ij = stress tensor

Substituting volume integrals gives,

Recognizing that this holds for any CV, the integral may be

droppedCauchys

Equation


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The Navier-Stokes Equation

As seen, Cauchy's equation is not useful in its present form, there are ten

(10) unknowns:

6 independent component of stress tensor (ij )

1 Density ()

3 independent Velocity components (V)

4 equations (continuity + momentum)

Thus in order to resolve flow field, 6 more equations are required to close

problem!

Thus,ij is separated into pressure and viscous stresses terms:

ijxx xy xz

yx yy yz

zx zy zz

p 0 00 p 0

0 0 p

xx xy xz

yx yy yz

zx zy zz

Viscous Stress Tensor

When fluid is at rest p acts & this p always

acts inward & normal to surface

When fluid is in motion p still acts inward &

but viscous stress may also exist


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Situation not yet improved, 6 unknowns in ij 6

unknowns in ij & 1 in P, which means one more is added!Reduction in the number of variables is achieved by relating

shear stress to strain-rate tensor (as it can be related tovelocity and material property).

For Newtonian fluid with constant properties (e.g. & T =

const.).

Substituting Newtonian closure into stress tensor gives:

Using the definition of ij :

Shear strain rate can be expressed in Cartesian coordinatesas:

, ,xx yy zzu v w

x y z

1 1 1, ,

2 2 2xy zx yz

u v w u v w

y x x z z y


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The Navier-Stokes Equation Combine linear strain rate and shear strain rate into one

symmetric second-order tensor called the strain-rate tensor.

Upon Substitution:

Substituting ij into Cauchys equation below gives us the N-S

eqns:

(6)


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Substitute equation (6) into three Cartesian coordinate of

Cauchy's equation (5), taking x-direction

Similar expression can be arrived for y and z directions.Also note that as long as velocity components are smooth

function, the order of differentiation is irrelevant, thus

This is from x-componentThis is from z-component

For Incompressible, the

term in parentheses =0


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Thus equation becomes .. the Navier Stokes eqns:

This results in a closed system of equations!

4 equations (continuity and momentum equations)

4 unknowns (U, V, W, p)

Note: Above is unsteady, nonlinear, second order partial differential eqn.

Incompressible NSE

written in vector formAdvectionPressure gradient Body force Diffusion


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Continuity

X-momentum

Y-momentum

Z-momentum


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In the field of CFD, the conservation of mass, momentum,(and/or energy) equations are collectively referred to as

the Navier-Stokes Equations.

The Navier-Stokes equations are a coupled set of non-

linear partial differential equations for five unknowns: r, u,v, w, E.

Additional thermodynamic relations (equation of state,

etc.) are needed to relate pressure and temperature to

other thermodynamic variables.

Transport properties (m and k) must also be specified.

May also be functions of pressure and temperature


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Note:

The N-S equations (inviscid flow) have been simplified tremendously; however, still cannot be

solved due to the nonlinear terms (i.e., uu/x, vu/y, wu/z, etc.).

Numerical methods such as the finite element and finite difference methods are often used to

approximate the fluid flow problems.

The Navier-Stokes EquationFor frictionless or inviscid flows ( 0, 0 & xx= yy =

zz = - p), the momentum equation reduces to Eulersequation:

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Summing all terms of heat added and dividing by dxdydz gives

the net rate of heat transfer to the fluid particle per unit volume:

Fouriers law of heat conduction relates the heat flux to the local

temperature gradient:

The total rate of work done by surface stresses can be

calculated, however for brevity we only consider x-components

of stresses.

Rate of work done (x-direction), normal =

Rate of work done (x+ x), normal =

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Energy Equation

qdivz

q

y

q

x

q zyx

z

Tkq

y

Tkq

x

Tkq zyx

zyu xx .

zyxx

uu xxxx

..


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Total of work done (x-direction), tangential =

Note that equation for stresses is similar to momentum term,

taking similar form:

Where vicious terms can be given as dissipation function , any

body forces by S & heat flux can be replaced by temperature

gradients, hence

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Energy Equation

z

w

y

v

x

u zxyxxy ...


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This far E has not been defined, but this E could be

defined as Internal energy, kinetic energy etc. Note SE is

source term.Example: For 2D incompressible flow, neglecting KE so h

can be reduced to CpT, also for most fluid engineering

problems local time derivative of p and dissipation function

is neglected, hence equation derived becomes:

Local accelerationAdvection Diffusion


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Energy Equation

Equation of State

),( TP

TCC

Tkk

T

pp

)(Property Relations

This far E has not been defined, but this E could bedefined as Internal energy, kinetic energy and gravitational

energy (body forces & includes effect of potential energy)


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Summary of Equations

The system of equations is now closed, with seven equations

for seven variables: pressure, three velocity components (u,v,

w), enthalpy, temperature, and density.

Under

Source term


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A Generic Form of Basic Equations

Note significant commonalitiesbetween the various equations.

Turbulence equations

(Not derived ) see it

takes same form.


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A Generic Form of Basic Equations

If Energy: specific heat generation

0

zyx

u wv1If mass:

If Vmomentum:V

u x

pS

1

T

k

qST

Using a general variable , the conservative form of all fluid

flow equations can usefully be written in the following form:


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Exact Solutions of the NSE

Solutions can also beclassified by type or

geometry

1. Couette shear flows

2. Steady duct/pipe flows

3. Flows with moving

boundaries

4. Similarity solutions

5. Asymptotic suction flows

6. Wind-driven Ekman flows

There are about 80

known exact solutions

to the NSE

The can be classified

as:

Linear solutions where

the convective

term is zero

Nonlinear solutionswhere convective term

is not zero


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Exact Solutions of the NSE

1. Set up the problem and geometry, identifying allrelevant dimensions and parameters

2. List all appropriate assumptions,

approximations, simplifications, and boundaryconditions

3. Simplify the differential equations as much aspossible

4. Integrate the equations5. Apply BC to solve for constants of integration

6. Verify results

Procedure for solving continuity and NSE


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Boundary ConditionsIn order to solve the Navier-Stokes equations, we must supply

appropriate boundary conditions/initial conditions for thefluid domain.

Boundary conditions at walls (solid surfaces)

fluid sticks to the walls (no-slip condition)

thermal boundary conditions at wallsconstant temperature

constant heat flux

others (convection, radiation, etc.)

Inflow/Outflow boundary conditions Where fluid enters the domain, appropriate inflow

conditions must be specified

velocity components total pressure and flow directionscalar variables e.g. temperature


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Boundary Conditions

Where fluid exits the domain, assumptions must bemade about flow conditions at the outlet

static pressure is knownflow is fully-developed

Other boundary conditions

SymmetryPeriodicFar-field


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Example exact solution

Fully Developed Couette Flow

For the given geometry and BCs, calculate the velocityand pressure fields, and estimate the shear force per

unit area acting on the bottom plate

Step 1: Geometry, dimensions, and properties

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Step 2: Assumptions and BCs

Assumptions

1. Plates are infinite in x and z

2. Flow is steady, /t = 0

3. Parallel flow, Vy=04. Incompressible, Newtonian, laminar, constant properties

5. No pressure gradient

6. 2D, W=0, /z = 0

7. Gravity acts in the -z direction,

Boundary conditions1. Bottom plate (y=0) : u=0, v=0, w=0

2. Top plate (y=h) : u=V, v=0, w=0

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Step 3: Simplify3 6

Note: these numbers referto the assumptions on the

previous slide

This means the flow is fully developedor not changing in the direction of flow

Continuity

X-momentum

2 Cont. 3 6 5 7 Cont. 6

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Step 4: Integrate

Z-momentum

X-momentumintegrate integrate

integrate

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Step 5: Apply BCs

y=0, u=0=C1(0) + C2 C2 = 0

y=h, u=V=C1h C1 = V/h

This gives

For pressure, no explicit BC, therefore C3 can remain

an arbitrary constant (recall only P appears in NSE).

Let p = p0 at z = 0 (C3 renamed p0)

1. Hydrostatic pressure

2. Pressure acts independently of flow

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Boundary conditions

Finally, calculate shear force on bottom plate

Shear force per unit area acting on the wall

Note that w is equal and opposite to the

shear stress acting on the fluid yx(Newtons third law).

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Steady 2D incompressible, laminar f low between two

stationary parallel infinite plates with height = H

Example

Assumpt ions:Assumpt ions:

AIRAIR (Working fluid), Laminar(Working flu id), Laminar

L = 1.0 mL = 1.0 m H = 0.1 mH = 0.1 m

U = 0.01 m/sU = 0.01 m/s = 1.2 kg/m= 1.2 kg/m33 airair

= 2 x 10= 2 x 10--5 kg/m5 kg/m--ss


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Steady 2D incompressible, laminar flow between

two stationary parallel plates with H= 0.1 m and

L=0.5m

(a) Velocity vector plot(a) Velocity vector plot

CFD simulationCFD simulation

resultsresults -- the flowthe flow

field along channelfield along channel

length changes fromlength changes from

uniform at inletuniform at inlet

surface to parabolicsurface to parabolic

profi le as i t travelsprofi le as i t travels

downstream.downstream.


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Results CFD

(b) U velocity contour plot(b) U velocity contour plot

(c) V velocity contour plot(c) V velocity contour plot

In hydrodynamic entranceIn hydrodynamic entrance

region (x


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Results CFD

(d) Static Temperature contour plot(d) Static Temperature contour plot

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Physical Interpretation Continuity Eqn.

Continuity equation applied to infinitesimal small control volume for

2D case of the fluid flow between two parallel stationary plates.

Consider , then u(x+ x) > u(x), since more fluid is

physically leaving the CV then entering along the x direction, there

should be more fluid entering than leaving y direction. Here

and the v (y+y) < v(y)

0x

u

0y

v

Next. , then u(x+ x) < u(x), since more fluid is

physically entering the CV then entering along the x direction,

there should be more fluid leaving than entering y direction. Here

and the v (y+ y) > v(y). (CONTINUTIY EQ.

SATISFIED)

0x

u

0y

v


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Turbulence and its modelling

All flows become unstable above a certain Reynolds

number.

At low Reynolds numbers flows are laminar.

For high Reynolds numbers flows are turbulent.

The transition occurs anywhere between 2000 and 1E6,depending on the flow.

For laminar flow problems, flows can be solved using the

conservation equations developed previously.

For turbulent flows, the computational effort involved insolving those for all time and length scales is prohibitive.

An engineering approach to calculate time-averaged flow

fields for turbulent flows will be developed.


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Examples of simple turbulent flows


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As air flows over and around objects in its

path, spiraling eddies, known as Von

Karman vortices, may form.

The vortices in this image were created

when prevailing winds sweeping east

across the northern Pacific Ocean

encountered Alaska's Aleutian Islands

Weddell sea in southern Atlantic

area near Antarctica,.


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Flow transitions around a cylinder

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Turbulence

. occurs at high Re

. are chaotic

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Turbulence

. are disspative

. are diffusive

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Turbulence ..

. rotation and

vorticity


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What is turbulence?


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Energy Cascade The continual transfer of energy from larger eddy to smaller and

smaller eddy is termed as Energy cascade .

Larger eddies highly anisotropic (varying in all directions),inertial effect dominates viscous effect.

Smaller eddies isotropic, viscous effect dominates and smears

out the directional ity of flow.Sources

Pope, Stephen B. Turbulent Flows.

Cambridge University Press

2000.

Tennekes H., Lumley J.L. A First Course

in Turbulence. The

MIT Press 1972.


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Turbulence


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Turbulence modelling objective


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Example: flow around a cylinder at Re=1E4

The figures show:

An experimental

snapshot.

Streamlines for time

averaged flow field.

Note the difference

between the time

averaged and the

instantaneous flow field.

Effective viscosityused to predict time

averaged flow field.


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Fortunately!

The Engineers are often interested in time average

properties of the flow (mean v, mean p, mean stress etc.)

Thus by adopting suitable time averaging operator details

concerning to the instantaneous fluctuations can be

discarded. To illustrate the above influences of turbulent fluctuations

on the mean flow, instantaneous continuity and Navier

Stokes equations for an incompressible flow with constant

viscosity is produced (time averaging governing

equations), this more popularly known as ReynoldsAverage Navier-Stokes (RANS) equations.


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To investigate the effects of fluctuations, we replace the

flow variables:


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Considering the x-direction, the time average x-direction

momentum equations terms becomes:

Considering the y & z direction , the time average momentumequations becomes:


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T b l d it d lli


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These terms are extra stresses terms ( 9 in all, 3 normal

stresses and 6 tangential stresses) and are called

REYNOLD STRESSES

REYNOLD STRESSES


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REYNOLD STRESSES

T b l d it d lli


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T b l l d lli


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Turbulence closure modelling

Turbulence models


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Turbulence models

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Turbulence models


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Turbulence models

Note Turbulence models are developed that close the

system of mean flow equations (Table presented earlier). In most engineering application the only the effects of

turbulence of mean flow is sought hence and details of

turbulent fluctuations are not resolved.These models uses

the Reynolds

AverageEquations and

forms the basis of

turbulence

calculations in

many CFD codes

Requires Time dependent flow equations solved

for mean flow and largest eddies and effects of

smaller eddies are modelled95

Turbulence models


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Turbulence models

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Turbulence models


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Turbulence models

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Boussinesq hypothesis


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Boussinesq hypothesis

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Turbulent viscosity


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Turbulent viscosity

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Turbulent Diffusivity


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Turbulent Diffusivity

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Predicting the turbulent viscosity


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Predicting the turbulent viscosity

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Mixing length model


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Mixing length model

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Mixing length model


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Mixing length model

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Spalart-Allmaras one-equation model


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Spalart Allmaras one equation model

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The k- model


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The k model

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Mean flow kinetic energy K


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Mean flow kinetic energy K

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Turbulent kinetic energy k


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Turbulent kinetic energy k

Model equation for k


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Model equation for k

108


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Dissipation rate - analytical equation


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ss pat o ate a a yt ca equat o

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Model equation for


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q

111

Calculating the Reynolds stresses from k &


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g y

112

k- model discussion


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More two-equation models


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q

114


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Improvement: RNG k-


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p

116

RNG k- equations


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q

117

Improvement: realizable k-


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p

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Realizable k- equations


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Realizable k- C equations


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Realizable k- positivity of normal stresses


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k- model


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Algebraic stress model


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Non-linear models


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Reynolds Stress Model


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Reynolds stress transport equation


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RSM equations


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RSM equations continued


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Setting boundary conditions


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Recommendation


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Lecture2_CFD_Course_Governing Equations [Compatibility Mode].pdf