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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

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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

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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

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Grid structure

Figure : Mesh structure of a two-dimensional staggered grid

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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.

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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

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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

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Interface velocity ui+1,j

ui+3/2,jui+1/2,j

xi+3/2xi+1/2 xi+1xi

yj

∆x

ui+1,j

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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

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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

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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 )

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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

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Plot of A(z)

Figure : Comparison of the velocity component u along the verticalcenterline of the cavity for Re = 100.

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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).

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Plot of F (z)

Figure : Comparison of the velocity component u along the verticalcenterline of the cavity for Re = 100.

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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)

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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)

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Iterative computation

• Linearization of the BVP

(u2)x → Uux

• Compute ui+1,j iteratively: update U and P etc.

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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.

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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.

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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.

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Numerical results

Flow in a lid driven cavity

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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.

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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)

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Flat plate boundary layer flow

Figure : Flow over a flat plate at zero incidence.

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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).

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Comparison with Blasius solution

Figure : Comparison of the function f ′(η) = u/U0 along a flat plate atzero incidence.

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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

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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).