lecture 16 lid driven cavity flow
DESCRIPTION
about the problem of Lid driven cavity flowTRANSCRIPT
Minor 2
Proposed date : Mar 27 (Fri) – Mar 30 (Mon).Please let me know now (latest by Friday) if this conflicts with any other Minors etc
Pattern : A few programs will be put up and you will be asked to make corrections, inferences, modifications etcWill also be split into basic, intermediate and advanced problems
Final Project
Default Project : Systematical computational analysis of boundary layer over a flat plateDue on Apr 28, last day of classAuditors
Those auditing the course officially need to either do a simple coding project or a literature review and answer a series of questions (your choice)Those auditing unofficially (just sitting in the class) obviously don’t need to do anything ☺
People taking the course for credit : Please stay back and a) Make a project group today and communicate it to me todayb) Choose on a project topic after discussing it with me today
Lid Driven Cavity flowFirst step of (all) incompressible NS based computational projects : An incompressible NS solver for Lid Driven Cavity flow
Staggered grid : x-momentum equation
The circled convective terms have to be found by interpolation as they don’t lie on the known grid points
This staggered grid formulation is also known as the Marker and Cell (MAC) formulation
Staggered grid : y-momentum equation
The circled convective terms have to be found by interpolation as they don’t lie on the known grid points
This staggered grid formulation is also known as the Marker and Cell (MAC) formulation
Now, we need to use the pressure Poisson equation to update the pressure
Discretization of the Pressure Poisson equation
The LHS is the usual 5-point finite difference for Poisson equationLeads to somewhat inaccurate transients because D is never really exactly zero.But, this method can be used for steady flows.
MAC algorithm for steady incompressible flows using the Pressure Poisson equation
Step 1 : Initialize u,vStep 2 : At each time step n:
Solve the Pressure Poisson equation to calculate pressure at level nUse the momentum equations, u, v and p at n to update the velocitiesSee if steady state criterion is reached to desired tolerance. If not, repeat Step 2.
A method for unsteady incompressible flows
The algorithm only satisfies the discretizedcontinuity equation approximatelyThere are also several other methods for steady incompressible flows. We’ll be discussing those when we deal with the finite volume methodFor an unsteady problem, we would like to ensure that the continuity equation is satisfied exactly (to machine precision) at each time step. There are several ways to do this. Let us try a small variation of the steady MAC method
MAC method for unsteady flow
Let us try to satisfy the continuity equation for each (i,j) at all time steps.
MAC method for unsteady flow
Momentum equations
At each time step we can updatethe velocities using the momentumequations.However, we need to do this in a waywhich will satisfy continuity as well
MAC method for unsteady flow
Momentum equations
At each time step we can updatethe velocities using the momentumequations.However, we need to do this in a waywhich will satisfy continuity as well
The way to do this is to retain the above discretizations but change the pressure discretizations to time level n+1
MAC method for unsteady flow
Momentum equations
At each time step we can updatethe velocities using the momentumequations.However, we need to do this in a waywhich will satisfy continuity as well
The way to do this is to retain the above discretizations but change the pressure discretizations to time level n+1
( )
( )21,
1,
11,
1
21,
,21
1,
1,1
1
,21
+
+++
+
+
+
+++
+
+
+−∆∆
−=
+−∆∆
−=
ji
nji
nji
n
ji
ji
nji
nji
n
ji
RHSVppytv
RHSUppxtu
MAC for unsteady, incompressible flow
Initialize solution for velocity. This may or may not satisfy the discrete continuity equationAt each time step
1. Solve the pressure equation
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
∆
−+
∆
−
∆=
∆
+−+
∆
+−
−+−+
++
++−
++
++−
y
VHSRVHSR
x
HSURHSUR
t
yppp
xppp
jijijiji
nji
nji
nji
nji
nji
nji
21,
21,,
21,
21
2
11,
1,
11,
2
1,1
1,
1,1
1
22
MAC for unsteady, incompressible flow (contd.)
At each time step 1. Solve the discrete Poisson equation
2. Update the velocities using⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
∆
−+
∆
−
∆=
∆
+−+
∆
+−
−+−+
++
++−
++
++−
y
VHSRVHSR
x
HSURHSUR
t
yppp
xppp
jijijiji
nji
nji
nji
nji
nji
nji
21,
21,,
21,
21
2
11,
1,
11,
2
1,1
1,
1,1
1
22
( )
( )21,
1,
11,
1
21,
,21
1,
1,1
1
,21
+
+++
+
+
+
+++
+
+
+−∆∆
−=
+−∆∆
−=
ji
nji
nji
n
ji
ji
nji
nji
n
ji
RHSVppytv
RHSUppxtu
MAC for unsteady, incompressible flow (contd.)
At each time step 1. Solve the pressure equation
2. Update the velocities using⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
∆
−+
∆
−
∆=
∆
+−+
∆
+−
−+−+
++
++−
++
++−
y
VHSRVHSR
x
HSURHSUR
t
yppp
xppp
jijijiji
nji
nji
nji
nji
nji
nji
21,
21,,
21,
21
2
11,
1,
11,
2
1,1
1,
1,1
1
22
( )
( )21,
1,
11,
1
21,
,21
1,
1,1
1
,21
+
+++
+
+
+
+++
+
+
+−∆∆
−=
+−∆∆
−=
ji
nji
nji
n
ji
ji
nji
nji
n
ji
RHSVppytv
RHSUppxtu
Note that using this methodensures that the discretecontinuity eqn is satisfiedexactly at each step
0
1
21,
1
21,
1
,21
1
,21
=∆
−+
∆
− +
−
+
+
+
−
+
+
y
vv
x
uu n
ji
n
ji
n
ji
n
ji
Summary of Lecture 15
MAC scheme for steady flows Explicit artificial compressibility method Pressure Poisson equation based method
MAC scheme for unsteady flowsDerive a Poisson equation after discretizing the flow equations This ensures that the discrete continuity equation is satisfied exactly
Plan for the remaining 12 lectures
FDM for non-Cartesian domains : 1.5 lecturesFinite Volume method : 5.5 lecturesSpectral methods : 1 lectureMultigrid methods : 1 lectureTurbulence modeling : 1 lectureHyperbolic conservation laws : 1 lectureLES and DNS : 1 lecture
Minor 2
Proposed date : Mar 27 (Fri) – Mar 30 (Mon).Please let me know now (latest by Friday) if this conflicts with any other Minors etc
Pattern : A few programs will be put up and you will be asked to make corrections, inferences, modifications etcWill also be split into basic, intermediate and advanced problems
Final Project
Default Project : Systematical computational analysis of boundary layer over a flat plateDue on Apr 28, last day of classAuditors
Those auditing the course officially need to either do a simple coding project or a literature review and answer a series of questions (your choice)Those auditing unofficially (just sitting in the class) obviously don’t need to do anything ☺
People taking the course for credit : Please stay back and a) Make a project group today and communicate it to me todayb) Choose on a project topic after discussing it with me today