Spatial discretization scheme for incompressible viscous flows · incompressible navier-stokes...
Transcript of Spatial discretization scheme for incompressible viscous flows · incompressible navier-stokes...
1/29
Introduction Local BVP method Results
Spatial discretization scheme for incompressibleviscous flows
N. Kumar
Supervisors: J.H.M. ten Thije Boonkkamp and B. Koren
CASA-day 2015
2/29
Introduction Local BVP method Results
Challenges in CFD
• Accuracy a primary concern with all CFD solvers
How to get higher accuracy?
* Using higher order methods – higher computational effort
* Using finer grids – significant increase in computational effort
* Designing better numerical schemes
3/29
Introduction Local BVP method Results
Incompressible viscous flow
• Flow governed by the incompressible Navier-Stokes equations:
ux + vy =0
ut + (u2)x + (uv)y =− px + ε(uxx + uyy )
vt + (uv)x + (v2)y =− py + ε(vxx + vyy )
(ε = 1/Re)
• Spatial discretization: Finite volume method on a uniformstaggered grid
4/29
Introduction Local BVP method Results
Grid structure
Figure : Mesh structure of a two-dimensional staggered grid
5/29
Introduction Local BVP method Results
Control volumes
opi ,j
ui+3/2,jui+1/2,j
vi ,j+1/2
xi+1xi
yj
yj+1 opi ,j+1
∆x
∆y
Ωv
Ωu
o
Figure : Control volumes for the spatial discretization of the momentumequations.
6/29
Introduction Local BVP method Results
Discretization of the convective term
• Discretizing :(u2)x + (uv)y
• Discretized convective term(Nu(u)
)i ,j
= ∆y(u2i+1,j − u2i ,j
)+
∆x(ui+1/2,j+1/2vi+1/2,j+1/2 − ui+1/2,j−1/2vi+1/2,j−1/2
)• Second order accurate FVM
7/29
Introduction Local BVP method Results
Computation of interface velocities
• Required interface velocities:
ui+1,j , vi+i/2,j+1/2, ui+1/2,j+1/2
• Commonly used techniques:
* Average value
* Upwind value
8/29
Introduction Local BVP method Results
Interface velocity ui+1,j
ui+3/2,jui+1/2,j
xi+3/2xi+1/2 xi+1xi
yj
∆x
ui+1,j
9/29
Introduction Local BVP method Results
Our research
• Computing interface velocities using local two-point boundaryvalue problems
• Pros:
* Interface velocity depending on the local Peclet number
* Higher accuracy (lower error constants)
• Cons:
* Higher computational effort
* Slower convergence of the solutions
10/29
Introduction Local BVP method Results
Computation of ui+1,j
• Solve the two-point local BVP :
((u2)x − εuxx) = −px − ((uv)y − εuyy ),
for x ∈ [xi+1/2,j , xi+3/2,j ] subject to the boundary conditions,
u(xi+1/2,j) = ui+1/2,j , u(xi+3/2,j) = ui+3/2,j
11/29
Introduction Local BVP method Results
Solution strategy
• Homogeneous case
(u2)x − εuxx = 0
• Including pressure gradient
(u2)x − εuxx = −px
• Including the pressure gradient and the cross flux term
(u2)x − εuxx = −px − ((uv)y − εuyy )
12/29
Introduction Local BVP method Results
The homogeneous case
• Further simplification:
- Linearize the equation-
Uux − εuxx = 0, (U is an estimate for ui+1,j)
• Solution -
uhi+1,j = A(−P/2)ui+1/2,j + A(P/2)ui+3/2,j
P ≡ U∆x
ε, A(z) ≡ (ez + 1)−1
13/29
Introduction Local BVP method Results
Plot of A(z)
Figure : Comparison of the velocity component u along the verticalcenterline of the cavity for Re = 100.
14/29
Introduction Local BVP method Results
Including the pressure gradient
Uux − εuxx = −px
• Assumption: p is piecewise linear over (xi+1/2,j , xi+3/2,j)
• ui+1,j as a sum of homogeneous and inhomogeneous part
ui+1,j = uhi+1,j + upi+1,j ,
upi+1,j = −(∆x)2
4ε
[F (−P/2)
pi+1 − pi∆x
+ F (P/2)pi+2 − pi+1
∆x
],
F (z) ≡ ez − 1− z
z2(ez + 1).
15/29
Introduction Local BVP method Results
Plot of F (z)
Figure : Comparison of the velocity component u along the verticalcenterline of the cavity for Re = 100.
16/29
Introduction Local BVP method Results
Including the cross flux term
Uux − εuxx = −px − ((uv)y − εuyy )︸ ︷︷ ︸constant = Ci+1,j
• Assumption: ((uv)y − εuyy ) is piecewise constant over(xi+1/2,j , xi+3/2,j)
ui+1,j = uhi+1,j + upi+1,j + uci+1,j
uci+1,j =1
ε∆x2
Ci+1,j
P(A(P/2)− 0.5)
17/29
Introduction Local BVP method Results
Interface velocity ui+1,j
ui+1,j = uhi+1,j + upi+1,j + uci+1,j
uhi+1,j = A(−P/2)ui+1/2,j + A(P/2)ui+3/2,j ,
upi+1,j = −(∆x)2
4ε
[F (−P/2)
pi+1 − pi∆x
+ F (P/2)pi+2 − pi+1
∆x
],
uci+1,j =1
ε∆x2
Ci+1,j
P(A(P/2)− 0.5)
18/29
Introduction Local BVP method Results
Iterative computation
• Linearization of the BVP
(u2)x → Uux
• Compute ui+1,j iteratively: update U and P etc.
19/29
Introduction Local BVP method Results
Interface velocities: ui+1/2,j+1/2 and vi+1/2,j+1/2
• For ui+1/2,j+1/2 use the local BVP
Vuy − εuyy = 0, yj < y < yj+1,
u(yj) = ui+1/2,j , u(yj+1) = ui+1/2,j+1,
• For vi+1/2,j+1/2 use
Uvx − εvxx = 0, xi < x < xi+1,
v(xi ) = vi ,j+1/2, v(xi+1) = vi+1,j+1/2.
20/29
Introduction Local BVP method Results
Interface velocity ui+1/2,j+1/2
ui+1/2,j+1
ui+1/2,j
xi
yj
∆yyj+1/2
yj+1
ui+1/2,j+1/2
Figure : Interface velocity ui+1/2,j+1/2.
21/29
Introduction Local BVP method Results
Interface velocity vi+1/2,j+1/2
xi+1/2 xi+1xi
yj
∆x
vi+1/2,j+1/2vi ,j+1/2 vi+1,j+1/2
Figure : Interface velocity vi+1/2,j+1/2.
22/29
Introduction Local BVP method Results
Numerical results
Flow in a lid driven cavity
23/29
Introduction Local BVP method Results
Validation for driven cavity flow
−0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Veloc ity component (u ) along the vert ical centerline
y
Ghia, Ghia (128 × 128)
1-D local BVP method (8 × 8)
Present (8 × 8)
1-D local BVP method (16 × 16)
Present (16 × 16)
1-D local BVP method (32 × 32)
Present (32 × 32)
Figure : Velocity component u along the vertical centerline of the cavityfor Re = 100.
24/29
Introduction Local BVP method Results
Flow in a lid driven cavity at Re = 100
−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
u
0.0
0.2
0.4
0.6
0.8
1.0
y
Ghia-Ghia-Shin (128 · 128)Standard average method
Upwind method1D local BVP method
Present method
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure : Comparison of the velocity component u along the verticalcenterline of the cavity for Re = 100. Grids used are 8× 8 (dotted lines),16× 16 (dashed lines) and 32× 32 (solid lines)
25/29
Introduction Local BVP method Results
Flat plate boundary layer flow
Figure : Flow over a flat plate at zero incidence.
26/29
Introduction Local BVP method Results
Re-sensitivity
Figure : Plot of velocity component u along the center of the plate overa family of Re = 4i × 100, (i = 0, 1, 2, 3, 4, 5).
27/29
Introduction Local BVP method Results
Comparison with Blasius solution
Figure : Comparison of the function f ′(η) = u/U0 along a flat plate atzero incidence.
28/29
Introduction Local BVP method Results
Conclusions
• Interface velocities dependent on Peclet number
- Average: P→ 0
- Upwind: P→∞
• Iterative computation: fast convergence
• Does not affect the formal order of accuracy, lower errorconstants
• Increased accuracy with the inclusion of pressure gradient andcross flux terms
29/29
Introduction Local BVP method Results
References
[1] N. Kumar, J.H.M. ten Thije Boonkkamp, and B. Koren. A newdiscretization method for the convective terms in theincompressible navier-stokes equations. In Finite Volumes forComplex Applications VII - Methods and Theoretical Aspects.
[2] N. Kumar, J.H.M. ten Thije Boonkkamp, and B. Koren. Asub-cell discretization method for the convective terms in theincompressible navier-stokes equations. In InternationalConference on Spectral and Higher Order Methods 2014(Submitted).