methods for unsteady problem

1

In computing unsteady flows, we have a fourth coordinate direction to consider: time

Just as with the space coordinates, time must be discretized

Essentially all unsteady solution methods advance in time in a step-by-step or “marching” manner

These methods are very similar to ones applied to initial value problems for ordinary differential equations (ODEs)

A brief review of such methods is given here

DR. AJMAL SHAH, PIEAS

CH-06: METHODS FOR UNSTEADY PROBLEMS

2

Method involving the values of unknown at only two timesExplicit or forward Euler methodImplicit or backward Euler methodMidpoint ruleTrapezoid ruleRichardson extrapolation

Two-Level Methods

Predictor-Corrector and Multipoint MethodsThe predictor-corrector methodRunge-Kutta methodsAdams-Bashforth methodsAdams-Moulton methodsLeapfrog methodCrank-Nicolson method


METHODS FOR INITIAL VALUE PROBLEMS IN ODES

3

)}(,{)( ttfdttd

Consider the first order ordinary differential equation with an initial condition:

ttt

tttttt

nn

o

1

12

1

.......

We have to find the solution ϕ a short time Δt after the initial point at to

11

)}(,{1n

n

n

n

t

t

nnt

t

dtttfdtdtd

oot )(

n

.......

2

1


TWO-LEVEL METHODS

4

rule) Trapezoid(

Midpoint rule provides the basis for Leapfrog method and

Trapezoid rule provide the basis for Crank-Nicolson method


TWO-LEVEL METHODSttf n

nnn ),(1 method)Euler forwardor Explicit (

ttf nn

nn

),( 11

1 method)Euler backwardor Implicit (

ttfn

n

nn

),( 21

21

1 rule)Midpoint (

ttftf nn

nn

nn

)],(),([21 1

11

5

There is a wide variety of predictor-corrector methods available

One, which is so well-known that it is called the predictor-corrector method (2nd order accurate)

In this method, the solution at the new time step is predicted using the explicit Euler method

The solution is then corrected by applying the trapezoid rule


PREDICTOR-CORRECTOR METHOD

Corrector)(;)],(),([21

Predictor)(;),(

1*

11

1*

ttftf

ttf

nnn

nnn

nn

nn

6

The second order Runge-Kutta method consists of two steps

The first may be regarded as a half-step predictor based on the explicit Euler method

It is followed by a midpoint rule corrector which makes the method second order

Corrector)();,(.

Predictor)();,(2

21*

21

1

21*

nn

nn

nn

nn

tft

tft


RUNGE-KUTTA METHODS

7

Runge-Kutta methods of higher order (fourth order) have been developed

The first two steps of this method use an explicit Euler predictor and an implicit Euler corrector

This is followed by a midpoint rule predictor for the full step and a Simpson's rule final corrector

corrector)Euler Implicit ();,(2

predictor)Euler Explicit ();,(2

21*

212

1**

21*

nn

nn

nn

nn

tft

tft


RUNGE-KUTTA METHODS

8

corrector) rule sSimpson'()];,(),(2

),(2),([6

predictor) ruleMidpoint ();,(.

corrector)Euler Implicit ();,(2

predictor)Euler Explicit ();,(2

1*

121**

21

21*

21

1

21**

211

*

21*

212

1**

21*

nnnn

nn

nn

nn

nn

nn

nn

nn

nn

nn

tftf

tftft

tft

tft

tft


RUNGE-KUTTA METHODS

9

qgraddivudivt

)()()(

11

)}(,{1n

n

n

n

t

t

nnt

t

dtttfdtdtd

)}(,{2

2

ttfxx

ut


APPLICATION TO THE GENERIC TRANSPORT EQUATION

)}(,{)( ttfdttd

In order to study properties of the explicit Euler and other simple schemes, we consider the 1D case of above Eq. with constant velocity, constant fluid properties, and no source terms

10

xtuc

xtd

Where

cdcdd

txx

u

ni

ni

ni

ni

ni

ni

ni

ni

nin

ini

;)(

,

)2

()2

()21(

)])(

2()2

([

2

111

211111

method)Euler forwardor Explicit (;),(

)}(,{

1

111

ttf

dtttfdtdtd

nn

nn

t

t

nnt

t

n

n

n

n


EXPLICIT EULER METHOD

)}(,{2

2

ttfxx

ut

11

xtuc

xtd

;

)( 2

The parameter d is the ratio of time step to the characteristic diffusion time

Which is roughly the time required for a disturbance to be transmitted by diffusion over a distance Δx

ni

ni

ni

ni

cdcdd 111 )

2()

2()21(



12

xtuc

xtd

;

)( 2

The second quantity c is the ratio of time step to the characteristic convection time, Δx/u,

The time required for a disturbance to be convected through a distance Δ x

This ratio is called Courant number and is one of the key parameters in computational fluid dynamics

ni

ni

ni

ni

cdcdd 111 )

2()

2()21(



13

The above Eqt. gives ϕ after a short time Δt

ni

ni

ni

ni

cdcdd 111 )

2()

2()21(



nn A 1

This equation gives the solution at the new time step in terms of the solution at the previous time step

14

ni

ni

ni

ni

ni

ni

ni

ni

nin

ini

cddcd

txx

u

11

11

1

2

111

11

11

111

)2

()2

()21(

)])(

2()2

([

11

)}(,{1n

n

n

n

t

t

nnt

t

dtttfdtdtd

In this method, all of the fluxes and source terms are evaluated in terms of the unknown variable values at the new time level

The result is a system of algebraic equations very similar to the one obtained for steady problems


IMPLICIT EULER METHOD

15

niPWEP

WE

PniW

niE

niP

tQ

tAAA

xxuA

xxuA

WhereQAAA

;)(

)(2;

)(2

,

22

11

11

1

The only difference lies in an additional contribution to the coefficient AP and to the source term QP

The above equation may be written as;


IMPLICIT EULER METHOD

16

So far we deal with generic conservation equationThe discretization principles described there also

apply to the similar terms in the Navier-Stokes equations

We shall now describe the treatment of the terms in the momentum equations which differ from those in the generic conservation equation

The unsteady and advection (convection) terms in the momentum equations have the same form as in the generic conservation equation


CH-07: SOLUTION OF THE NAVIER-STOKES EQUATIONS

iij

ij

j

iji gxp

xxuu

tu

)()(

17

The diffusive (viscous) terms are similar to their counterparts in the generic equation but, because the momentum equations are vector equations, these contributions become a bit more complex

The momentum equations also contain a contribution from the pressure, which has no analog in the generic equation

It may be regarded either as a source term or as a surface force

But, due to the close connection of the pressure and the continuity equation, it requires special attention


SOLUTION OF THE NAVIER-STOKES EQUATIONS

iij

ij

j

iji gxp

xxuu

tu

)()(

18

The treatment of the convective term in the momentum equations follows that of the convective term in the generic equation

Any of the previous methods (Ch:04) can be used

j

ij

xuu

)(

The convective term in the momentum equation is non-linear; its differential and integral forms are:

iij

ij

j

iji gxp

xxuu

tu

)()(


DISCRETIZATION OF CONVECTIVE AND VISCOUS TERMS

S

i ndSvu .

19

j

ij

x

The viscous terms in the momentum equations correspond to the diffusive term in the generic equation

Their differential and integral forms are:

iij

ij

j

iji gxp

xxuu

tu

)()(



S

jij ndSi ).(

20

)(j

i

j xu

x

For Newtonian incompressible flows the viscous stresses become;

)(i

j

j

iij x

uxu

This term can be discretized using any of the approaches described previously for generic equation



The viscous term is complicated than the generic diffusive term

The part of the viscous term in the momentum equations (Vector) which corresponds to the diffusive term in the generic conservation equation is

S

i ndSgradu ).(

21

But it is only one contribution of viscous effects to the ith component of momentum

The contributions due to the bulk viscosity are non-zero only in compressible flows

) The contributions due to the spatial variability of the viscosity are zero for incompressible flows with constant properties



)(j

i

j xu

x

)(

i

j

j

iij x

uxu

S

i ndSgradu ).(

)(i

j

j xu

x

Sj

i

j ndSixu

).(

22

In FV methods, the pressure term is usually treated as a surface force (conservative approach), i.e. in the equation for ui the following integral is used

iij

ij

j

iji gxp

xxuu

tu

)()(

S

i ndSpi .

Alternatively, the pressure can be treated non-conservatively, by retaining the above integral in its volumetric form:

digradp i.



digradp i.

23

Solution of the Navier-Stokes equations is complicated due to unavailability of independent pressure equation

But, pressure gradient contributes to each of the three momentum equations

On the other hand, the continuity equation does not have a dominant variable in incompressible flows

Mass conservation is a kinematic constraint on the velocity field rather than a dynamic equation


CALCULATION OF THE PRESSURE

24

One way out of this difficulty is to construct pressure field based on satisfying the continuity equation

The absolute pressure is of no significance in an incompressible flow

Only the gradient of the pressure is importantIn compressible flows the continuity equation is used to

calculate densityPressure is calculated from equation of state


CALCULATION OF THE PRESSURE

iij

ij

j

iji gxp

xxuu

tu

)()(

25

The above equation can be written as given below, the spatial derivatives are not important here so we have lumped them in to a single term H

Now apply the explicit Euler method

i

nni

ni

ni x

pHtuu )()( 1


A SIMPLE EXPLICIT TIME ADVANCE SCHEME

ii

j

ij

ij

iji

xpH

xxp

xuu

tu

)()(

iij

ij

j

iji gxp

xxuu

tu

)()(

26

The above equation can be used to calculate the velocity at new time step, but this will need the pressure gradient

But this pressure gradient must satisfy continuityTo find such pressure gradient, let us take the gradient

of the above equation

i

nni

ni

ni x

pHtuu )()( 1



i

nni

ii

ni

i

ni

xpH

xt

xu

xu )()( 1

27

The above equations must enforce the continuity, and the equation obtained will be the required pressure equation (Poisson equation)

The velocity obtained with this pressure will satisfy continuity

If pressure is treated implicitly, we will have pn+1, instead of pn



i

ni

i

n

i xH

xp

x

0)( 1

i

ni

xu 0)(

i

ni

xu

28

This provides the following algorithm for time-advancing the Navier-Stokes equationsi. Start with a velocity field ui

n at time tn which is assumed divergence free (if it is not divergence free this can be corrected)

ii. Compute the combination, Hin , of the advective

and viscous terms and its divergenceiii. Solve the Poisson equation for the pressure pn

iv. Compute the velocity field at the new time step (it will be divergence free)

v. The stage is now set for the next time step



29

Semi-Implicit Method for Pressure-Linked Equations.1. Guess the pressure field P*.2. Solve the u, v and w momentum equations to calculate u*,

v* and w*.3. Solve the pressure correction equation for 4. Calculate P from; 5. Calculate u, v and w from u*, v* and w* using their

correction formulas.6. Solve the discretized equations of other ϕ’s if they

influence the flow field through fluid properties, source terms etc.

7. Treat the corrected pressure P as a new guessed P*, return to step 2 and repeat the whole procedure till convergence.

PPPP *


THE SIMPLE ALGORITHM

methods for unsteady problem

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