Computational Fluid Solving the Dynamics Navier-Stokes ...gtryggva/CFD-Course2017/Lecture-15-2017.pdfComputational Fluid Dynamics Colocated grids Computational Fluid Dynamics Colocated

Computational Fluid Dynamics


http://www.nd.edu/~gtryggva/CFD-Course/

Grétar Tryggvason

Lecture 15March 20, 2017


Solving theNavier-StokesEquations in

Primitive Variables

http://www.nd.edu/~gtryggva/CFD-Course/


The projection method for incompressible flows—revisited

Colocated grids

Methods for the incompressible Navier-Stokes Equations Moin and Kim Bell, et al Artificial compressibility

Methods for all speeds

OutlineComputational Fluid Dynamics

∇h ⋅ui , jn +1 = 0

Discretization in time

�

ui, jn +1 − ui, j

n

Δt= −A i, j

n − ∇hPi, j +Di, jn

Summary of discrete vector equations

No explicit equation for the pressure!

Evolution of the velocity

Constraint on velocity


ui , jn +1 − ui , j

t

Δt= −∇hPi , j ⇒ ui , j

n +1 = ui, jt − Δt ∇hPi, j

ui , jt − ui, j

n

Δt= −Ai, j

n +Di, jn ⇒ ui, j

t = ui, jn + Δt −Ai, j

n + Di, jn( )

�

ui, jn +1 − ui, j

n

Δt= −A i, j

n − ∇hPi, j +Di, jn

Split

by introducing the temporary velocity ut

into

and

Projection Method

Discretization in timeComputational Fluid Dynamics

�

ui, jn +1 = ui, j

t − Δt ∇ hPi, j

∇h ⋅ui , jn +1 = 0

∇h ⋅ui , jn +1 =∇h ⋅ui, j

t − Δt ∇h ⋅∇hPi, j0

To derive an equation for the pressure we take the divergence of

and use the mass conservation equation

∇h2Pi, j =

1Δt

∇h ⋅ui, jt

The result is

Discretization in time


ui , jn +1 = ui , j

t − Δt ∇hPi, j

ui , jt = ui, j

n + Δt −Ai , jn +Di, j

n( )

∇h2Pi, j =

1Δt

∇h ⋅ui, jt

1. Find a temporary velocity using the advection and the diffusion terms only:

2. Find the pressure needed to make the velocity field incompressible

3. Correct the velocity by adding the pressure gradient:

Discretization in timeComputational Fluid Dynamics

ui+1/2,j-1 pi+1,j-1pi,j-1

vi,j-3/2

ui+1/2,j+1

ui+1/2,j

pi+1,j+1pi,j+1

vi,j+3/2

pi+1,jpi,j

vi,j+1/2

vi,j-1/2

vi-1,j-3/2

vi-1,j+3/2

vi-1,j+1/2

vi-1,j-1/2

vi-1,j-3/2

vi-1,j+3/2

vi-1,j+1/2

vi-1,j-1/2

pi-1,j-1

pi-1,j+1

pi-1,j

ui-1/2,j-1

ui-1/2,j+1

ui-1/2,j

ui+1/2,j-1

ui+1/2,j+1

ui+1/2,j

ui+1/2,j-1

ui+1/2,j+1

ui+1/2,j

The Staggered (MAC) Grid

vi,j+1/2

Pi,ju i-1/2,j ui+1/2,j

vi,j-1/2

v - cell

u - cell

The Staggered

Grid


Velocity and pressure Velocity and vorticity


Colocated grids


Colocated gridsAlthough staggered grids have been very successful, in some cases it is desirable to use co-located (or colocated) grids where all variables are located at the same physical point.


vi,j+1/2

Pi,jui-1/2,j ui+1/2,j

vn

uw ue

vsvi,j-1/2

Pi,j,ui,j,vi,j

Colocated gridsStaggered grids

All variables are stored at the same location


First idea: use averaging for the variables on the edges:�

ui, j* = ui, j − ΔtA(ui, j

n )

ui, jn+1 = ui, j

* − Δth(pe − pw )

uen+1 − uw

n+1 + vnn+1 − vs

n+1 = 0

�

pe =12pi+1, j + pi, j( )

�

ui, jn+1 = ui, j

* − Δth12pi+1, j + pi, j( ) − 12 pi, j + pi−1, j( )

⎛ ⎝ ⎜

⎞ ⎠ ⎟

�

ue =12ui+1, j + ui, j( )

�

ui, jn+1 = ui, j

* − Δt2h

pi+1, j − pi−1, j( )

Split equations for the u velocity


Substituting

�

ui, jn +1 = ui, j

* −Δt2h

pi+1, j − pi, j−1( )into

�

uen +1 − uw

n+1 + vnn +1 − vs

n +1 = 0

yields

�

pi+2, j + pi− 2, j + pi, j+2 + pi, j− 2 − 4 pi, j =2hΔt

ui+1, j* − ui−1, j

* + vi, j+1* −vi, j−1

*( )

�

ue =12ui+1, j + ui, j( ) and


The pressure points are also uncoupled and the pressure field can develop oscillations

A straight forward application discretization on colocated grids results in a very wide stencil for the pressure


The remedy is to find the pressures that make the edge velocities incompressible

Rhie and Chow. AIAA Journal. 21 (1983), 1525-1532.


vn

uw ue

vs

Pi,j,ui,j,vi,j

Instead of interpolating (the final velocity)

�

uen +1 = ue

* −Δth

pi+1, j − pi, j( )

�

ue* =12ui+1, j* + ui, j

*( )

�

uen +1 =

12ui+1, jn +1 + ui, j

n+1( )

interpolate (the intermediate velocity)

and then find

In effect, “pretend” we are using a staggered grid!

The Rhie and Chow method


�

uen +1 = ue

* −Δth

pi+1, j − pi, j( )

�

uen +1 − uw

n+1 + vnn +1 − vs

n +1 = 0

�

ue* =12ui+1, j* + ui, j

*( )

�

ue* −Δth

pi+1, j − pi, j( )⎛ ⎝ ⎞ ⎠ − uw* −

Δth

pi, j − pi−1, j( )⎛ ⎝ ⎞ ⎠

+ vn* −Δth

pi, j+1 − pi, j( )⎛ ⎝ ⎞ ⎠ − vs

* − Δth

pi, j − pi, j−1( )⎛ ⎝ ⎞ ⎠ = 0

�

pi+1, j + pi−1, j + pi, j+1 + pi, j−1 − 4 pi, j =hΔt

ue* − uw

* + vn* − vs

*( )Rearrange:

Substitute

whereinto

giving


Substituting for the velocities:

�

pi+1, j + pi−1, j + pi, j+1 + pi, j−1 − 4 pi, j =h2Δt

ui+1, j* − ui−1, j

* + vi, j+1* −vi, j−1

*( )

�

ui, jn +1 = ui, j

* −Δt2h

pi+1, j − pi−1, j( )

�

pe =12pi+1, j + pi, j( )

For the correction of the momentum equation we still use the average of the pressures

giving


The boundary conditions for the velocity are now very simple: The velocity at nodes on the wall is simply the wall velocity.

The pressure boundary is more complex:


Find the pressure gradient by applying the Navier-Stokes equations at a point at the boundary

�

ρ∂v∂t

+ u∂v∂x

+ v∂v∂y

⎛ ⎝ ⎜ ⎞

⎠ = −

∂p∂y

+ µ∂ 2v∂x 2

+∂ 2v∂y2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

At the wall, most of the terms are zero

�

∂p∂y

= µ∂ 2v∂y 2

�

pi,2 − pi,1h

= µ∂ 2v∂y 2

⎞ ⎠ ⎟ i,1

Evaluated by one-sided differences

�

i,1

�

i,2

�

pi,2 − pi,1h

= µ vi,1n − 2vi,2

n + vi,3n

h2


Define:

We can write:

�

i,1

�

i,2

�

pi,2 − pi,1h

= µ vi,1n − 2vi,2

n + vi,3n

h2

�

pi,2 − pi,1 =hΔtvi,1*

�

vi,1* = Δt

hpi,2 − pi,1( ) = µΔth2 vi,1

n − 2vi,2n + vi,3

n( )


Write the pressure equation for j=2

�

pi,2 − pi,1 =hΔtvi,1*

�

pi+1,2 + pi−1,2 + pi,3 + pi,1 − 4 pi,2 =h2Δt

ui+1,2* − ui−1,2

* + vi,3* −vi,1

*( )

And use

For the pressure at j=1


The algorithm is therefore:

1. First find predicted velocities:

�

ui, j* = ui, j − ΔtA(u

n )

2. Find pressure by solving:

�

pi+1, j + pi−1, j + pi, j+1 + pi, j−1 − 4 pi, j =h2Δt

ui+1, j* − ui−1, j

* + vi, j+1* −vi, j−1

*( )

3. Correct the velocities:

and

�

ui, jn +1 = ui, j

* −Δt2h

pi+1, j − pi−1, j( ) and

suitably modified at the boundaries

�

vi, j* = vi, j − ΔtA(v

n )

�

vi, jn+1 = vi, j

* −Δt2h

pi, j+1 − pi. j−1( )


% projection method in primitive variables on a colocated mesh nx=19;ny=19;dt=0.0025;nstep=300;mu=0.1;maxit=100;beta=1.2;h=1/nx; time=0; u=zeros(nx,ny);v=zeros(nx,ny);p=zeros(nx,ny); ut=zeros(nx,ny);vt=zeros(nx,ny); u(2:nx-1,ny)=1.0; for is=1:nstep for i=2:nx-1,for j=2:ny-1 ut(i,j)=u(i,j)+dt*(-(0.5/h)*(u(i,j)*(u(i+1,j)-u(i-1,j))+v(i,j)*(u(i,j+1)-u(i,j-1)))+(mu/h^2)*(u(i+1,j)+u(i-1,j)+u(i,j+1)+u(i,j-1)-4*u(i,j))); vt(i,j)=v(i,j)+dt*(-(0.5/h)*(u(i,j)*(v(i+1,j)-v(i-1,j))+v(i,j)*(v(i,j+1)-v(i,j-1)))+ (mu/h^2)*(v(i+1,j)+v(i-1,j)+v(i,j+1)+v(i,j-1)-4*v(i,j))); end,end vt(2:nx-1,1)=(mu*dt/h^2)*(v(2:nx-1,3)-2*v(2:nx-1,2)); vt(2:nx-1,ny)=(mu*dt/h^2)*(v(2:nx-1,ny-2)-2*v(2:nx-1,ny-1)); ut(1,2:ny-1)=(mu*dt/h^2)*(u(3,2:ny-1)-2*u(2,2:ny-1)); ut(nx,2:ny-1)=(mu*dt/h^2)*(u(nx-2,2:ny-1)-2*u(nx-1,2:ny-1)); for it=1:maxit % solving for pressure p(2:nx-1,1)=p(2:nx-1,2)+(h/dt)*vt(2:nx-1,1); p(2:nx-1,ny)=p(2:nx-1,ny-1)-(h/dt)*vt(2:nx-1,ny); p(1,2:ny-1)=p(2,2:ny-1)+(h/dt)*ut(1,2:ny-1); p(nx,2:ny-1)=p(nx-1,2:ny-1)-(h/dt)*ut(nx,2:ny-1); for i=2:nx-1,for j=2:ny-1 p(i,j)=beta*0.25*(p(i+1,j)+p(i-1,j)+p(i,j+1)+p(i,j-1)- (0.5*h/dt)*(ut(i+1,j)-ut(i-1,j)+vt(i,j+1)-vt(i,j-1)))+(1-beta)*p(i,j); end,end

p(floor(nx/2),floor(ny/2))=0.0; % set the pressure in the center. Needed since bc is not incorporated into eq end u(2:nx-1,2:ny-1)=ut(2:nx-1,2:ny-1)-(0.5*dt/h)*(p(3:nx,2:ny-1)-p(1:nx-2,2:ny-1)); v(2:nx-1,2:ny-1)=vt(2:nx-1,2:ny-1)-(0.5*dt/h)*(p(2:nx-1,3:ny)-p(2:nx-1,1:ny-2)); time=time+dt hold off,quiver(flipud(rot90(u)),flipud(rot90(v)),'r'); hold on;axis([1 nx 1 ny]);axis square,pause(0.01) end plot(10*u(floor(ny/2),1:nx)+floor(nx/2),1:nx),hold on; plot([1,nx],[floor(ny/2),floor(ny/2)]),plot([floor(nx/2),floor(nx/2)],[1,ny]) plot(10*v(1:nx,floor(ny/2))+floor(ny/2));axis square,axis([1,nx, 1,ny])


Colocated grids

2 4 6 8 10 12 14 16 18

2

4

6

8

10

12

14

16

18

Re=10


Why colocated grids:Sometimes simpler for body fitted gridsEasy to use methods for hyperbolic equationsEasier to implement AMRSome people just don’t like staggered grids!


The two-dimensional programs developed in the project and shown here can be extended to fully three-dimensional flows in a relatively straight forward way, replacing u(i,j) by u(i,j,k), etc. The time required to run the code increases significantly and visualizing the output becomes more challenging.


Higher Order in Time


Forward in time, centered in space (summary):

�

u* −u n

Δt= −A u n( ) + ν∇2u n

∇2p = 1Δt

∇ ⋅u*

u n+1 = u* −Δt∇p

�

U 2 =max(u2 + v 2)

Time step limitations

where

�

Δtadv ≤2νU 2

& Δtdiff ≤14h2

ν

�

τ = LU

�

Δt '= Δtτ

≤ 2νU 2

UL

= 2Re

�

Δt '= Δtτ

≤ 14h2

νUL

= 14

hL

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2

Re

Therefore the nondimensional time step is:

And Δt à 0 for Re à 0 and Re à ∞

and

Define


Forward in time, centered in space:

�

Δt '= 2Re

�

Δt '= 14hL

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2

Re

What is the maximum timestep?

Δt à 0 for Re à 0 and Re à ∞

and

Increasing h

h à 0 Re

Δt’

�

Δt 'max =12hL�

ReΔt ' max = 8Lh


Second order in time—Predictor-Corrector

�

u* −u n

Δt= −∇u nu n + ν∇2u n

∇2p* = 1Δt

∇ ⋅u*

u1 = u* −Δt∇p*

�

u ** − u1

Δt= −∇u1u1+ν∇2u1

∇2p** = 1Δt

∇⋅u **

u 2 = u ** − Δt∇p**

�

u n+1 = 12u n + u 2( )

Then average the results

Backward step using the predicted velocity:

A second order method can be developed by first taking a forward step, then a backward step and average the results:


Advanced Solvers

For low Re, use implicit methods for diffusion termFor high Re, use stable advection schemes

Combine both for schemes intended for all Re


Other Methods


Fully Implicit

�

un +1 − un

Δt=12

− A (un ) +A(un +1)( ) +ν ∇h2un + ∇ h2un+1( )[ ] −∇p∇ ⋅un+1 = 0

Solve by iteration

Rarely used due to the complications of the nonlinear system that must be solved for the advection terms


Adam-Bashford/Crank-Nicolson

�

un +1 − un

Δt= −

32A(un ) − 1

2A(un−1)⎛ ⎝

⎞ ⎠ +

ν2

∇h2un + ∇ h

2un+1( ) −∇p∇ ⋅un+1 = 0

�

˜ u −un

Δt= − 3

2A (un ) − 1

2A (un−1)⎛ ⎝

⎞ ⎠ +

ν2∇ h

2un

un +1 - ˜ u Δt

= −∇p + ν2∇ h

2un+1

∇ ⋅un+1 = 0

�

∇ h2p = 1

Δt∇ ⋅ ˜ u

Split:

The correction equation is implicit and must be solved by an iteration in the same way as the pressure equation


Method of Kim and Moin (JCP 59 (1985), 8-23)

�

˜ u −un

Δt= − 3

2A (un ) − 1

2A (un−1)⎛ ⎝

⎞ ⎠ +

ν2

∇ h2un + ∇h2 ˜ u ( )

un +1 − ˜ u Δt

= −∇φ

∇ ⋅un+1 = 0

�

∇ h2φ =

1Δt

∇⋅ ˜ u

The first equation is implicit and must be solved by an iteration in the same way as the pressure equation


�

un +1 − un

Δt= −

32

A(un ) − 12

A(un−1)⎛ ⎝ ⎞ ⎠ +

ν2

∇h2un + ∇ h

2un+1( ) + ν2 ∇h2 ˜ u −∇ h

2un+1( ) −∇φ

�

ν2∇h

2 ˜ u −∇ h2un+1( ) −∇φ = ν2∇ h

2φ −∇φ = ∇p

Where we have added and subtracted an implicit diffusion term.

�

un +1 − ˜ u Δt

= −∇φ

we can rewrite the last terms as:

�

∇p

Using

Notice that Φ is not exactly p. Adding the first two equations gives


Method of Bell, Colella and Glaz (JCP 85 (1989), 7-83)

�

un +1 − un

Δt= −A (un+1/ 2) + ν

2∇ h2un + ∇h

2un +1( ) − ∇p∇ ⋅un+1 = 0

A Godunov method is used for the advection terms


The Artificial Compressibility Method (Chorin, 1967, JCP 2, 12-26)

∂u∂t

= −A(u) + ν∇2u − ∇p

∂ p∂t

=−1ε∇ ⋅u

ε is a numerical parameter, in effect an artificial velocity of sound. At steady state

�

∇ ⋅u = 0

∂u∂t

= −A(u) + ν∇2u − ∇p − 12∇ ⋅u( )u

Sometimes we keep additional terms

∂u∂t

= −A(u) + ν∇2u − ∇p − 12∇ ⋅u( )u − 1

ε∇ ∇ ⋅u( )


Many other methods have been proposed

SIMPLE (Semi-Implicit Method for Pressure-Linked Equations). One of the earliest method for steady state problems. Many extensions.

PISO (Pressure Implicit with Split Operator). Similar to the projection method but iterates to enforce incompressibility


The SIMPLE (Semi-Implicit Method for Pressure Linked Equations) family of methods for the steady state solution

At steady state we have

0 = −A(u) + ν∇2u − ∇p∇ ⋅u = 0

apu p = alulneighbours∑ −∇ph

∇ ⋅u = 0

After discretization the momentum equation can be written as:

Where the sum is over nonlinear terms since the coefficients depend on the velocity


In SIMPLE based methods we solve the discrete momentum equation

approximately using an estimation of the pressure and then correct it by adding pressure.

Since the momentum equation is an approximation, this is not the final velocity and we therefore continue iterating until convergence

apu p = alul

neighbours∑ −∇ph

p = p* + p 'u = u* + u '

∇2 p ' = − 1

Δt∇ ⋅u p

*


The SIMPLE Algorithm:

1.  Guess the pressure field p*

2.  Solve the momentum equations to get u* and v* (using u to find ap)

3.  Solve for p’

4.  Compute new pressure

5.  Compute new velocity

6.  Take the corrected pressure as the new guess p* and go to step 2, using the new velocity for the coefficient

u p = u p

* − ∇ph

p = p* + p '

∇2 p ' = − 1

Δt∇ ⋅u*

apu p

* = alul*

neighbours∑ −∇ph*

In the original SIMPLE method we drop this term


There is no good reason for neglecting the neighboring points in step 2 when updating the velocities (except that they are not known!) and in the SIMPLEC (SIMPLE Corrected) the neighboring velocities are estimated using the values from the last iteration.

In SIMPLER (SIMPLE Revisited) method pressure is found from the new velocity.

In the PISO (Pressure Implicit with Splitting of Operators) method the first estimate for the velocities is followed by another one where the preliminary velocities are used to get a better estimate.


Higher orderin time for the pressure-

velocity formulation


�

u * − u n12 Δt

= −∇u nu n +ν∇2u n

∇2p* = 2Δt

∇⋅u *

u1 = u * − Δt2∇p*

�

u ** − u n12 Δt

= −∇u1u1+ν∇2u1

∇2p** = 2Δt

∇⋅u **

u 2 = u ** − Δt2∇p**

continue for the second step:

First a half step:A complete Runge-Kutta time integration (Weinan E.)


�

u *** − u n

Δt= −∇u 2u 2 +ν∇2u 2

∇2p*** = 1Δt

∇⋅u ***

u 3 = u *** − Δt∇p***

�

k = Δt −∇u 3u 3 +ν∇2u 3( )

�

u+ = 13 −un + u1+ 2u 2 + u 3( ) + 16 k

∇2p+ = 1Δt

∇⋅u+

u n+1 = u+ − Δt∇p+

And finally

Then compute

Take a full step using the predicted velocity

A complete Runge-Kutta time integration (continued)


�

u * − u n12 Δt

= −∇u nu n +ν∇2u n

u ** − u *12 Δt

= −∇u *u * +ν∇2u *

u *** − u n

Δt= −∇u *u * +ν∇2u *

u+ = 13 −un + u * + 2u ** + u ***( ) + Δt6 −∇u

***u *** +ν∇2u ***( )

∇2p+ = 1Δt

∇⋅u+

u n+1 = u+ − Δt∇p+

Simplified Fourth order method


Boundary conditionsInflow and outflow


Boundary conditions

Fully enclosed flowDriven cavity

Internal flow (inflow & outflow)

External flowFlow over bodies


For fully enclosed domains, such as in the driven cavity problem, the application of the boundary conditions is relatively straight forward.

For domains with inflows it is usually reasonable to specify the velocity profile at the inlet

The major problem is how to handle outflow boundaries in such a way that the fluid leaves in a “physically plausible” way


The Backward Facing Step Problem

Velocity given

�

u = u(y)v = 0

�

∂u∂x

= 0

�

∂v∂x

= 0


u

vnx+1,j

P

unx,j Pnx,j

u P

v

u

unx+1,j

u

vnx+2,j

vv(nx+2,1:ny+1)=v(nx+1,1:ny+1);

u(nx+1,j)=u(nx,j);

p(nx,j)=(1/2)*(p(nx,j+1)+p(nx,j-1)-(h/dt)*(v(nx+1,j+1)-v(nx+1,j)));

�

∂u∂x

= 0

�

∂v∂x

= 0

For the pressure, obtain an equation by applying conservation of mass to the control volume next to the boundary


Example


Simplified Backward Facing Step Problem

Velocity given

�

u = u(y)v = 0

�

∂u∂x

= 0

Velocity gradient given

�

∂v∂x

= 0


0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

Velocity


5 10 15 20 25 30

2

4

6

8

10

12

14

16

18

Velocity+vorticity


05

1015

20

0

10

20

30

40-2.5

-2

-1.5

-1

-0.5

0

0.5

Pressure

Inflow


Periodic Boundary conditionsIn many cases it is possible to use periodic boundary conditions, where what flows out through one boundary reappears flowing in through the opposite boundary. Such conditions are particularly suitable for theoretical studies of idealized flows. For such boundaries it is easiest to specify the pressure drop, but by adjusting the pressure gradient it is possible to specify the volume flux


External flow

Free outflow

Free stream (uniform flow, for example)

Boundary of computational domain


Other ways to deal with free-stream boundaries

Include potential flow perturbationCompute flow from vorticity distributionMap the boundary at infinity to a finite distance

Fundamentally, the specification of the boundary conditions does not have a unique solution and is also faced by experimentalists. However, by taking the boundaries far away and checking the solution for the effect of moving the boundaries, good results can be obtained


For compressible flow the challenges are even greater since reflected waves can propagate upstream. Two main approaches are in use:

Non-reflecting boundary conditions where characteristics or the linearized equations are used to specify the boundary state, and

Absorption or damping layers where the solution is smoothed out gradually to minimize reflected waves


The Advection

Terms


The advection terms can be written in several different forms

∇⋅uu = u ⋅∇u = 12u ⋅∇u+ 1

2∇⋅uu

=12∇ u ⋅u( )+ ∇×u( )×u = 12∇ u ⋅u( )+w×u

∂ρu∂t

+∇⋅ρuu = ρ ∂u∂t+u ⋅∇u

⎛

⎝⎜

⎞

⎠⎟= ρ

∂u∂t+∇⋅uu

⎛

⎝⎜

⎞

⎠⎟

Using mass conservation

Using incompressibility

w =∇×u

For incompressible flows we can rewrite the nonlinear term in several ways:

Most forms seem to give comparable results

Here


Methods for all speeds


Compressible low speed flows pose a particular challenge since the sound speed is very fast and the time step limitations can therefore be very severe. A number of methods have been proposed to handle both incompressible and compressible flowsMost of them use the energy equation in terms of the time derivative of the pressure


Yabe’s Method ∂ρ∂t+∇⋅ρu = 0

∂u∂t+u ⋅∇u = −1

ρ∇p

∂p∂t+u ⋅∇p = −ρc2∇⋅u

ρ* − ρn

Δt= −un ⋅∇ρn

u* −un

Δt= −un ⋅∇un

p* − pn

Δt= −un ⋅∇pn

ρn+1 − ρ*

Δt= −ρ*∇⋅un+1

un+1 −u*

Δt=−1ρ*∇pn+1

pn+1 − p*

Δt= −ρ*c2∇⋅un+1

Split the equations

Seong Y. Yoon and T. Yabe. The unified simulation for incompressible and compressible flow by the predictor-corrector scheme based on the CIP method. Computer Physics Communications 119 (1999) 149-158

Use the governing equations in non-conservative from

and


ρ* − ρn


u* −un

Δt= −un ⋅∇un

p* − pn

Δt= −un ⋅∇pn

ρn+1 − ρ*

Δt= −ρ*∇⋅un+1

un+1 −u*

Δt=−1ρ*∇pn+1

pn+1 − p*

Δt= −ρ*c2∇⋅un+1

!u−u*

Δt=−1ρ*∇p*

δ p = pn+1 − p*

un+1 − !uΔt

=−1ρ*

∇pn+1 −∇p*( ) = −∇δ pρ*

Split momentum again

and

whereTake divergence of the correction equation

∇⋅ un+1 − !u = −Δt ∇δ pρ*

⎛

⎝⎜

⎞

⎠⎟

and


∇⋅ un+1 − !u = −Δt ∇δ pρ*

⎛

⎝⎜

⎞

⎠⎟ ∇⋅un+1 −∇⋅ !u = −Δt ∇⋅ ∇δ p

ρ*⎛

⎝⎜

⎞

⎠⎟

pn+1 − p*

Δt= −ρ*c2∇⋅un+1 ∇⋅un+1 = − p

n+1 − p*

ρ*c2Δt= −

δ pρ*c2Δt

−δ p

ρ*c2Δt−∇⋅ !u = −Δt ∇⋅ ∇δ p

ρ*⎛

⎝⎜

⎞

⎠⎟

∇⋅∇δ pρ*

⎛

⎝⎜

⎞

⎠⎟−

δ pρ*c2Δt

=∇⋅ !u

Rewrite the pressure equation

substitute

or

or

Rearrange:


ρ* = ρn −Δtun ⋅∇ρn

u* = un −Δtun ⋅∇un

p* = pn −Δtun ⋅∇pn

∇⋅∇δ pρ*

⎛

⎝⎜

⎞

⎠⎟−

δ pρ*c2Δt

=∇⋅ !u

ρn+1 = ρ* −Δt ρ*∇⋅un+1

un+1 = !u− Δtρ*∇δ p

pn+1 = p* +δ p

The steps are

Predictor step

Solve pressure equation

Corrector step

!u = u* − Δtρ*∇p*Preliminary velocity


Kwatra’s Method ∂ρ∂t+∇⋅ρu = 0

∂ρu∂t

+∇⋅ρuu = −∇p

∂E∂t

+∇⋅Eu = −∇⋅ pu

Use the governing equations in conservative form

Nipun Kwatra, Jonathan Su, Jo ́n T. Gre ́tarsson, and Ronald Fedkiw. A method for avoiding the acoustic time step restriction in compressible flow. Journal of Computational Physics 228 (2009) 4146–4161

∂p∂t+u ⋅∇p = −ρc2∇⋅u

Supplemented by the pressure equations


ρn+1 − ρn


ρn+1un+1 − ρn+1u*

Δt= −∇p

En+1 − E*

Δt= −∇⋅ pun

ρn+1u* − ρnun

Δt= −∇ρnunun

E* − En

Δt= −∇⋅Enun

un+1 = u* − Δtρn+1

∇p

Start by finding the new density

Then split the momentum and energy equation

and

The corrected velocity is given by


∂p∂t+u ⋅∇p = −ρc2∇⋅u

∇⋅un+1 =∇⋅u* −Δt∇⋅ ∇pρn+1⎛

⎝⎜

⎞

⎠⎟

∇⋅u ≈ ∇⋅un+1

∂p∂t+u ⋅∇p = ρc2∇⋅u* −Δtρc2∇⋅ ∇p

ρn+1⎛

⎝⎜

⎞

⎠⎟

pn+1 − pn

Δt+u ⋅∇p = ρc2∇⋅u* −Δtρc2∇⋅ ∇p

n+1

ρn+1⎛

⎝⎜

⎞

⎠⎟

The divergence of the corrected velocity is:

Take in the pressure equation

Use the pressure equation to eliminate the divergence at n+1

Approximate the time derivative by an Euler step


The steps are:

ρn+1 = ρn −Δt un ⋅∇ρn

u* = ρn

ρn+1un − Δt

ρn+1∇ρnunun

E* = En −Δt ∇⋅Enun

Predictor step

Δtρc2∇⋅ ∇pn+1

ρn+1⎛

⎝⎜

⎞

⎠⎟+pn+1 − pn

Δt= −u ⋅∇p+ ρc2∇⋅u*Pressure equation

un+1 = u* − Δtρn+1

∇p

En+1 = E* −Δt ∇⋅ punCorrector equation


Several other methods have been proposed for all speed flows, based on similar considerations

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