An Immersed Boundary Method To Solve Flow And Heat Transfer Problems Involvin...
NIACFDS2015-09-29_HiroNishikawa_HNS20
1. Third-Order Edge-Based Scheme and
New Hyperbolic Navier-Stokes System
Hiroaki Nishikawa
National Institute of Aerospace
!
!
64th NIA CFD Seminar
September 29, 2015
Supported by Army Research Office (ARO), Software CRADLE, NASA
Towards efficient, accurate, robust 3rd-order unstructured CFD
2. Approaches to
Efficient and Accurate CFD
- Efficient and accurate discretization
- Efficient iterative solver
- Grid generation/adaptation
- High performance computing
- Efficient and accurate discretization
!
!
- Efficient iterative solver:
- Grid generation/adaptation
- High performance computing
3. 1. Third-Order Discretization that ends well.
2. Navier-Stokes Discretization that ends well.
If things turn out well, everything is well.
5. Third-Order Discretization That Ends Well
!
!
Exact for quadratic solutions and fluxes.
Then, all is well no matter how strange it seems.
Third-order accurate on unstructured grids.
7. 1st-Order
2nd-Order(Linear LSQ gradients)
k
j
nr
jk
nℓ
jk
Edge-Based DiscretizationEdge-Based Discretization
Order of Accuracy
3rd-Order (Quadratic LSQ gradients)
Katz&Sankaran(JCP2011)
Efficient 3rd-order scheme: edge-loop with a flux per edge
Only on simplex elements (triangles/tetrahedra).
8. Exact for Quadratic? Part I
Zero dissipation for quadratic solution
Linear extrapolation with quadratic LSQ gradients:
For a quadratic solution u, they both reduce to
The same left and right states, but not exact.
k
j
nr
jk
nℓ
jk
Quadratic exactness depends on the averaged flux term.
JCP2015, v281, pp518-555
j k
9. Exact for quadratic fluxes
Linear flux extrapolation with quadratic LSQ gradients:
and the edge-based discretization is exact for div(f):
The same left and right fluxes, but not exact.
k
j
nr
jk
nℓ
jk
Edge-based discretization ends well - Third-order
True for arbitrary triangles/tetrahedra
Exact for Quadratic? Part II
JCP2015, v281, pp518-555
j k
For a quadratic flux, it gives
10. All’s Well for Edge-Based Discretization
DO NOT use quadratic flux extrapolation, or lose 3rd-order
⭕️
❌
DO NOT use curved elements, or lose 3rd-order
Immediately applicable to existing grids
Note: Accurate surface normals needed
at boundary nodes for some BCs.
From CAD or by surface reconstruction.
Strange? But then the discretization is exact for quadratic solutions and fluxes.
See JCP2015, v281, pp518-555,
NIA CFD Seminar 12-16-2014
See JCP2015, v281, pp518-555,
NIA CFD Seminar 12-16-2014
All’s well, that ends well.
Confirmed for NS computations in AIAA2015-2451
See also JCP2015, 300, pp.455-491
11. Extensions toViscous Terms
Cubic LSQ gradients for viscous terms.
Second-derivatives for unsteady terms (source terms).
High-order curved grids required.
Or if we can write the viscous terms as a hyperbolic conservation law:
!
!
then the third-order scheme directly applies to the viscous terms.
Extended by Pincock and Katz, JSC, v61, Issue2, pp454-476
(See also JCP2012 for source terms)
Not straightforward due to compatibility requirement:
See JCP2014, NIA CFD Seminar 06-18-2013
13. Navier-Stokes Discretization That Ends Well
Consistent with the Navier-Stokes Equations
Then, all is well no matter how strange it seems.
We’re solving the NS equations.
14. Hyperbolic Method Since JCP2007
JCP2012JCP2007NASA-TM2014 In review
Hyperbolic Conservation Law
Website: hiroakinishikawa.com/fohsm
AIAA 2011-3043Extended to the compressible Navier-Stokes in 2011:
Note: 3rd-order edge-based scheme directly applies without modifications.
15. Advantages of Hyperbolic Method
1. Simple and Efficient Discretization
3. Improved Convergence
2. Accurate Gradients:
- Methods for hyperbolic systems directly apply.
- 1st-order diffusion/viscous scheme with1st-order gradients
(consistent Jacobian, P0-DG, etc.)
Reconstructed, LSQ (uy) Hyperbolic Method(uy)
See Mazaheri and Nishikawa,
JCP2015, 300, pp.455-491
A hyperbolic adv-diff solver available at cfdbooks.com
- Same order for solution and gradients
- Smooth gradients on irregular grids
- Stiffness due to high-order derivatives eliminated: O(1/h) speedup for diffusion.
- Time-to-solution = O(N^p) is reduced with a lower p. N = # of unknowns.
- Systematic/robust solver with 1st-order scheme: implicit solver, p-multigrid
16. Hyperbolic Method: Development
Diffusion - JCP2007
Advection Diffusion - JCP2010
Compressible Navier-Stokes - AIAA2011
Source terms - JCP2012
Time-dependent problems - NASA2014, CF2014 with Alireza Mazaheri
Incompressible Navier-Stokes - AIAA2014
Dispersion - In review, with Alireza Mazaheri (NASA), Mario Ricchiuto (INRIA)
!
3rd-order RD scheme - CF2014, AIAA2015, JCP2015, with Alireza Mazaheri
3rd-order Active-Flux scheme - AIAA2014/2015, with P. L. Roe (Michigan)
3rd-order EB scheme - JCP2012/2014/2015,AIAA2014/2015, with ARO, Cradle
Beyond 3rd-order: DG underway.
17. Steady diffusion equation First-order system (‘mixed form’)
Elliptic
Equivalent
It is much more straightforward to discretize
the first-order hyperbolic system than others.
Hyperbolic
First-orderParabolic
Hyperbolic
Hyperbolic Diffusion System
19. First-Order Formulation
A popular formulation (DG, FOSLS, etc.), often called a mixed form.
NOTE: The NS system is not a hyperbolic system.
The Navier-Stokes System:
Viscous terms
20. Navier-Stokes discretization that ends well
Navier-Stokes equations:
Consistent Discretization (Residual):
Leading term is the Navier-Stokes equations.
21. Hyperbolic Navier-Stokes System: HNS14
Just add pseudo-time terms:
Hyperbolic in pseudo time for both inviscid and viscous.
We write the system as
AIAA 2011-3043
AIAA 2014-2901
Free parameters:
22. Viscous Part is Hyperbolic
Viscous Jacobian has real eigenvalues:
Navier-Stokes Equations = Hyperbolic Inviscid + HyperbolicViscous
AIAA 2011-3043
Viscous and heating waves
x
t
23. Hyperbolic Navier-Stokes Discretization
HNS14:
Pseudo steady state or simply ignore , we have
Hyperbolic method is designed to end well:
Consistent discretization of the NS equations.
which is consistent with the Navier-Stokes equations:
Discretize as a hyperbolic system (e.g., Upwind FV, FEM, RD, etc.):
24. Common Approach to Euler Discretization
Hyperbolic/Elliptic in space - Steady Euler equations:
- Acoustic system is Cauchy-Riemann (Laplace eqs.) for subsonic flows.
- Space marching is not possible in subsonic flows.
Hyperbolic in time - Unsteady Euler equations:
Spatial discretization is typically constructed based on the latter, not the former,
e.g., Riemann solvers, Roe flux, SUPG, etc.: Almost all based on unsteady characteristics.
Hyperbolic method does the same to the viscous terms.
AIAA 2003-3704
25. Two Approaches to NS Discretization
The Navier-Stokes Equations:
Hyperbolize
Consistent NS Discretization:
O(1/h) speed-up higher-order/quality gradients
Hyperbolic approach
E.g., Upwind, FV, RD, etc.
Discretize
Conventional approach
Can we discretize NS without going
through hyperbolic forms, and arrive at
spatial discretization with similar features?
RGV-Approach (ICOSAHOM2014) by Harold Atkins
Hybridized-DG?
27. HNS14: Limitations
1. Reduced order of accuracy in velocity gradients(AIAA2014).
2. Scheme-II not possible: use accurate gradients for reconstruction
3. Second-derivatives required for third-order accuracy.
wish to avoid for both source terms and physical time derivatives.
⭕️❌❌
[Accurate gradients desired also for RD schemes and Active-flux schemes.]
AIAA2014-2901
29. HNS17: Hyperbolic Navier-Stokes 17
Spatial part (terms in black) is consistent with the Navier-Stokes equations
Replace viscous stresses by velocity gradients
AIAA2015-2451
30. HNS17: Not Good Enough…
Density gradient is required for SchemeII (and desired for RD/AF)…
How can we introduce the density gradient?
⭕️❌ ⭕️
HNS17
AIAA2015-2451
Brenner's modification
Not widely accepted yet……..
: Mass diffusion added to continuity eq. - We then hyperbolize it.
31. HNS20: Hyperbolic Navier-Stokes 20
Add “artificial hyperbolic diffusion” to HNS17
Small coefficient
< TE
Negligibly small
AIAA2015-2451
See JCP2014 , NIA CFD Seminar 06-18-2013
Scheme-II can be constructed.
A desired target system also for other schemes: e.g., RD/AF schemes.
33. Numerical Flux
Any inviscid flux can be employed: Roe’s flux is used here.
Inviscid flux
Viscous flux
AHD flux
AIAA2015-2451
Dissipation for preconditioned PDE: See AIAA 2003-3704
34. HNS20: Ultimate Hyperbolic NS System
Two formulations are equivalent: No approximations
Second derivatives are not needed in the discretization.
Original Formulation -1st/2nd-order schemes
Fully Hyperbolic Formulation - 3rd-order scheme
See AIAA2015-2451 for details.
AIAA2015-2451
35. Discretization of HNS20 (3rd-order)
Edge-based discretization: k
j
nr
jk
nℓ
jk
Source fluxes
AIAA2015-2451
36. Numerical Flux: Third-Order
Dissipation term plays a critical role for accuracy.
Upwind fluxes
applied to all
source terms
written as
hyperbolic
systems.
AIAA2015-2451
37. HNS20 Discretization Ends Well
Edge-based discretization:
Pseudo-steady state or simply ignore , we have
which consistently approximates the Navier-Stokes equations:
Navier-Stokes Equations
Hyperbolic discretization ends well: All is well.
39. One order lower for uy and vx…
HNS17/20: AccuracyVerification by MMS
Note: Hyperbolic formulation does not guarantee the same order of accuracy in
solution and gradients. Seemingly due to symmetry of the stress tensor.
40. k
j
nr
jk
nℓ
jk
Decoupling leads to Loss of Accuracy
Central flux
p decoupled >>> explicit Green-Gauss formula.
Reconstructed gradients are one order lower accurate than solution.
Globally coupled. Cannot solve locally for pj The strong coupling is
introduced by the dissipation term - Critical to achieve high accuracy.
Upwind flux
See JCP2014
41. Artificial Hyperbolic Dissipation
Hyperbolic Diffusion
Upwind Flux
Variables are now coupled as if they are solving Laplace equations.
Add only the dissipation
Laplacian
AIAA2015-2451
43. Second-Order HNS20, Scheme-II
Left state: Linear LSQ gradients
Left flux:
Inviscid: Viscous:
Left fluxes:
- 2nd-order gradients (r, g, q) by 2nd-order HNS20.
- 3rd-order inviscid scheme: truly 3rd-order for infinite Re.
- Based on 2nd-order algorithm: quadratic LSQ not needed.
Note: Inviscid flux depends only on
AIAA2015-2451
45. Jacobian-Free Newton-Krylov Solver
Consistent residual-Jacobian: Exact derivative of 1st-order scheme
Pseudo time derivative dropped: It is solving the NS equations.
Update
Newton-Krylov (GCR)
Variable Preconditioner: Implicit Defect-Correction solver
NOTE: Typically,1st-order viscous scheme is not available, and so,
0-th order scheme is employed for Jacobian, leading to
potential convergence deterioration.
AIAA2015-2451
55. Results: Flow over a Cylinder
Hyperbolic solvers still
converge faster on
finer grids:
!
Slopes are different.
Get 3rd-order solution
faster than conventional
2nd-order on the same
grid - Impossible, ususally.
AIAA2015-2451
57. Flat Plate: Velocity Profiles
Improvements observed in the transverse velocity.
68x48 grid
AIAA2015-2451
58. Flat Plate: Grid Convergence of Cd
2nd-order HNS
2nd-order Traditional NS
2nd-order viscous stress
1st-order viscous stress
2nd-order hyperbolic scheme is like a 3rd-order scheme.
Finer grids
548x388
34x24
AIAA2015-2451
59. Flat Plate: CPU Time vs Nodes
CPUTime
Again, hyperbolic solver converges faster for finer grids.
AIAA2015-2451
60. Flat Plate: Convergence History
Iteration CPUTime
Hyperbolic solver converges with a much fewer number of iterations.
AIAA2015-2451
61. Conclusions
- New systems proposed: HNS17, HNS20
- 2nd-order HNS20-II: 3rd-order inviscid + 2nd-order viscous
- Artificial hyperbolic dissipation - accurate velocity gradients
- Artificial hyperbolic diffusion - accurate density gradient
Future work:
- High-aspect ratio issue (not an issue for implicit solvers)
- Weak/Strong BCs
- 3rd-order unsteady scheme without computing second-derivatives
- Accurate gradients of primitive variables (HNS20)
62. Current Focus
Implementation into NASA’s FUN3D code
“3D Hyperbolic Navier-Stokes solver ends well”
to be demonstrated.
- Second-order HNS20-II
!
[ 3rd-order inviscid and 2nd-order gradients by 2nd-order scheme ]
!
- 3rd-order Navier-Stokes on tetrahedral grids
!
[ Keys: High-order surface normals in 3D, robust quadratic LSQ in 3D ]
With Dr.Yi Liu (NIA)
