**Developing CFD code and post-processors for **

**Developing CFD code and post-processors for**

**shock wave/turbulent boundary layer interactions **

**shock wave/turbulent boundary layer interactions**

### Stephan Priebe

### Research Associate

### University of Maryland

### Funded by the US Air Force Office of Scientific Research

### General Electric Global Research Technical Presentation

### Niskayuna, NY

### June 18, 2013

### •

### Classical problem in fluid mechanics:

* ‘What mechanism drives the characteristic low-frequency *

* unsteadiness in shock wave/turbulent boundary layer *

* interactions, and how can it be controlled?’ *

### •

### CFD codes to address this problem:

- Finite-volume code for steady, laminar simulations - DNS code for turbulent simulations

- LES code for turbulent simulations

### •

### Examples of post-processors to address this problem:

- Coherence estimation and associated error bounds

- Dynamic mode decomposition (DMD, also known as Koopman analysis)

### •

### DNS of a hypersonic shock interaction at Mach 7.2:

- Code validation

- Observations regarding turbulence model development and validation

• **S**hock Wave/**T**urbulent **B**oundary **L**ayer **I**nteractions (**STBLI**s) are common in
compressible flow applications.

• STBLIs are usually associated with: • Large-scale unsteady flow separation.

• Peak values of mean and fluctuating wall pressure and heat transfer.

• Non-uniform flow downstream of interaction.

STBLIs can cause damage to the vehicle surface and adversely affect engine performance.

**Introduction **

**Over-expanded**

**rocket nozzle**

**(Bourgoing &**

**Reijasse 2005).****Schematic of scramjet flow (Courtesy: R. Baurle, NASA)****. **

• Canonical STBLI configurations used in experiments and numerical simulations: • Compression ramp

• Reflected shock

• Sharp fin

• The flow dynamics are similar, regardless of the configuration. See Dussauge et
al. (2006); Piponniau et al (2009).
**Nominally ****two-dimensional ****three-dimensional ****flow**
**flow**
**flow**

**Introduction **

**Physical Description of Supersonic STBLIs**

• Parameters that govern the flow: Re, M, T_{w}/T_{r}, α.
• As the interaction strength α is increased, the flow

topology changes and mean-flow separation develops: • Small α: attached flow.

• Intermediate α: incipiently-separated flow.

• Large α: separated flow.

**wall **

• Consider supersonic STBLIs in which the flow reattaches downstream.

**Compression ramp experiments at M=2.85, Re**_{δ}**=1.7****×****10****6 **_{(Settles et al. 1979). }

**Ramp angle **

**α**

**Physical Description of Supersonic STBLIs**

• Characteristic low-frequency shock unsteadiness in separated STBLIs.

• The frequency of the unsteadiness is St = fL_{sep}/U_{∞}= 0.02-0.05 (Dussauge et al.
2006). This frequency is about two orders of magnitude lower than U_{∞}/δ, which is
the characteristic frequency of the turbulence in the incoming boundary layer.

x/Lref y/ L re f ρ/ρref

• Conservative, shock-capturing scheme required for simulating compressible flows with shock waves.

• Developed and validated 2D finite-volume code:

– Upwind scheme with Steger-Warming flux vector splitting

– Explicit time integration (forward Euler and 3rd-order Runge-Kutta)

– Implicit time integration (LU-SGS by Yoon & Jameson, AIAAJ 1988)

**CFD Codes and Simulations **

**Ex. 1: Circular Cylinder in Mach 5 flow****Ex. 2: Viscous flow over flat plate at Mach 2.5**

**Convergence rate of **

**implicit and explicit schemes**

y/

L

re

f

x/Lref

**Density contours: ****ρ****/****ρ****ref=0.5 (blue) ****to ****ρ****/****ρ****ref=1.25 (red) **

**DNS**

**CFD Codes and Simulations **

• If the flow is turbulent, there are conflicting requirements for shock-capturing and for high accuracy to resolve the turbulence.

• DNS code solves full three-dimensional unsteady Navier–Stokes equations in conservation form.

• Numerical method:

– Inviscid fluxes: 4th-order linearly and nonlinearly optimized WENO – Viscous fluxes: 4th-order central differences

– Time integration: 3rd-order, low-storage Runge-Kutta method

**Mean-wall pressure along compression ****ramp from DNS and experiment. **

• DNS code validated for compressible turbulent flows, including shock/turbulence interactions:

– Martin et al JCP 2006 – Martin JFM 2007

– Wu & Martin AIAAJ 2007 – Priebe et al. AIAAJ 2009

fi+1/2

x

**WENO Scheme **

Original weighted essentially non-oscillatory scheme by Jiang & Shu JCP 1996

**candidate stencils: **

**numerical flux: **

**smoothness measurement: **

Linear optimization

Weirs & Candler AIAA 1997

Martin *et al.* JCP 2006

Nonlinear optimization

Taylor *et al.* JCP 2007

• Numerical flux:

• Symmetric collection of candidate stencils:

• Weights in flux calculation:

• Limiters to reduce WENO adaptation:

**WENO Scheme **

Linear Optimization Weirs & Candler AIAA 1997

Martin *et al.* JCP 2006

**absolute limiter **

**relative limiter **

Nonlinear Optimization

Taylor *et al.* JCP 2007

Analytic curvilinear grids

**Grid, Boundary & Initial Conditions **

Boundary conditions

Initialization (Martin 2007)

• Separation of time scales is key challenge for numerics:

• Code is (mildly) unstable, which manifests itself for N_{iter} > 106.

• Example of numerical instability: Continuation of DNS run of Priebe et al. (2009)

**Modifications to CFD Code **

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**Movie of numerical Schlieren on a streamwise-wall normal plane. DNS of 12° reflected shock ****interaction at Mach 2.9 and Re**_{θ}** = 2400. **

• Inflow BC likely source of numerical instability

• Two types of code modifications to address the numerical instability: 1. ‘Fix-type’ modifications

2. Modifications that target source of numerical artefact

• Example of fix-type modification: ‘freestream-filter’ algorithm (Priebe & Martin 2009)

**Modifications to CFD Code **

**Schematic of rescaling-recycling inflow boundary condition. ****The method of Xu & Martin (2004) is used here. **

**Schematic of filter region****Filter function depending on computational **

**coordinate k:**

* Validation:* Code modifications enable long runtimes required to capture the shock

unsteadiness while maintaining accuracy and numerical stability (Priebe & Martin 2009).

**Modifications to CFD Code **

**DNS of Mach 2.9, Re****θ**** = 2900 boundary layer. Mass flux signal **

**in the freestream at z/****δ**** = 1.6. **

**Filter function depending on computational ****coordinate k:**

**Filtering operation on a conserved variable u:****no filtering **

**with filtering **

**Van Driest-transformed mean velocity profile **

**Modifications to CFD Code **

• Extending above arguments to compressible flow (Priebe & Martin 2012): • The appropriate internal velocity scale is according to Morkovin’s

hypothesis.

• Therefore, the Lagrangian length scale becomes:

• To eliminate spurious correlations in the computational box, the required rescaling length is O(30δ) for the present Mach 3 conditions, and O(40δ) for the Mach 7 conditions.

• Rescaling length is 8δ in Mach 3 DNS and 52δ in Mach 7 DNS.

• But, these code modifications do not address the actual source of the numerical
artefact. It turns out that the artefact stems from the computational setup. **We ****need to ‘rewrite the user manual’ for the code. **

• Importance of the streamwise rescaling length:

• Large-scale eddies take a significantly longer time to decorrelate with themselves as

they are being convected by the mean velocity than suggested by the length scale of
the autocorrelation function, that is: L_{Euler} << L_{Lagrange}.

• For incompressible flow, Simens et al. (2009) argue that a large eddy of size O(δ),

internal velocity O(u_{τ}) and convection velocity O(U_{∞}) will decorrelate with itself as it
convects over a distance O(U_{∞}δ/u_{τ}).

**Effect of Rescaling Inflow BC **

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### Mach 7.2 boundary layer DNS (Priebe & Martin AIAA 2011)

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### Mach 7.2 boundary layer DNS (Priebe & Martin AIAA 2011)

**Effect of Rescaling Inflow BC **

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### Mach 7.2 boundary layer DNS (Priebe & Martin AIAA 2011)

**Effect of Rescaling Inflow BC **

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### Mach 7.2 boundary layer DNS (Priebe & Martin AIAA 2011)

**Effect of Rescaling Inflow BC **

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• Proposed Lagrangian scaling

• Provides good estimate for Lagrangian eddy decorrelation length – at least in particular flow considered here.

• Relevant length scale to be used in rescaling-recycling algorithms for prescribing inflow BCs.

**Effect of Rescaling Inflow BC **

* R_{uu} vs. x for various time delays. *
• Traditional criterion for choosing

rescaling length (based on Eulerian decorrelation length) insufficient.

**R**_{uu}** vs. x at zero time delay. ****From Wu & Martin (2007). **

• Implemented LES capability in the CFD code (joint work with L. Duan, N. Grube, J. Li).

• Numerical method:

– Favre-filtered Navier-Stokes equations are solved

– Subgrid-scale stress and heat flux are closed with the dynamic mixed model

(Martin, Piomelli & Candler 2000)

Sample LES code:

– SGS subroutine called inside the main code

– SGS module and subroutine

**Large Eddy Simulations **

link to code here

• Dissipation added by WENO scheme indicated by nonlinearity index (NI), which lies between 0 and 1:

**Large Eddy Simulations – Code Validation **

**Li, Grube, Priebe & Martin (AIAA 2013) **

**no limiters ** **absolute **

**limiter **

High-quality LES solution obtained at greatly reduced cost compared to DNS.

**Large Eddy Simulations – Code Validation **

* Li, Priebe, Grube & Martin (upcoming AIAA paper June 2013) *
LES (blue line)

DNS (black line)

**DNS ** **LES ** **Filter width **

Nx 1024 288 4

Ny 160 40 4

Nz 128 64 2

Total 21M 0.7M 32

**Wall pressure distribution through the interaction **

• Classification of post-processors for unsteady flows:

• Two examples of high-level post-processors will be described here:

**1.**** *** Coherence (level 6)*, which may be thought of as a frequency-decomposed correlation
coefficient and provides an exercise in error estimation.

**2.**** *** Dynamic Mode Decomposition (level 7)*, which is a modal decomposition tool to
reveal the dominant flow dynamics.

Adapted from Deck (AIAA 2013)

**Developing Post-Processors**

1. Integrated quantities (aerodynamic forces & moments) 2. Mean quantities

3. Higher-order statistical moments (e.g. rms-fluctuations) 4. Cross-correlations

5. One-point spectral quantities (power spectral density) 6. Two-point spectral quantities (coherence, phase) 7. ‘Advanced’ analysis (e.g. modal decompositions)

Time-resolved data not required

Time-resolved data required

• * Definition:* Coherence between two continuous-time signals x(t) and y(t)

where is the power spectral density of x, f is frequency.

• * Definition:* Estimated coherence between two discrete time signals x(t

_{n}), y(t

_{n})

• * Algorithm:* Standard spectral estimation algorithm to determine the individual
spectral densities that appear in the estimated coherence:

1. Poor (noisy) algorithm:

2. Better (less noisy) algorithm:

• Algorithm 1 is the so-called periodogram; algorithm 2 is Welch’s method.

**Coherence Analysis**

coherence of 0.25?

** Example of noisy coherence plot obtained from data **

• Note: A coherence value of 0.25 is “equivalent” to a correlation coefficient of

√0.25 = 0.5, which would indicate a strong statistical link between the signals. • Physically, this would imply that the incoming boundary layer plays an important

role in driving the shock motion at low frequencies.

**Coherence Analysis**

Estimated coherence from DNS between the shock motion in the freestream and the pressure signal at two locations in the upstream, undisturbed boundary layer. The spectral estimator used is Welch’s method with Hamming window

• Regardless of the spectral estimator used (periodogram, Welch, or another

algorithm), the coherence estimate will have some level of bias error and variance.

• **Key observation:**

Both bias and noise increase as the parameter n is reduced. In other words, the bias & noise will be much more significant when short-duration numerical data is used as compared to the typically much longer experimental data.

**Coherence Analysis**

** Bias (left) and variance (right) of coherence estimate as a function of true coherence ****and n (number of non-overlapped segments). Carter, Knapp & Nuttall (1973). **

γ2 _{γ}2

n n

Bias

Variance

0 1 0 1

** The statistical properties of ****the coherence estimate ****must be known to draw **

**reliable inferences about the ****true coherence. **

• **Monte Carlo-type algorithm **to determine the statistical properties of the coherence

estimator (Welch’s method). See also e.g. Benignus (1969); Bortel & Sovka (2007).

• **Key challenge: **test signals must be an accurate representation of actual turbulence signals

in terms of distribution, autocorrelation, spanwise dependence.
• **Example:** to model the spanwise flow

dependence, an autoregressive model of order p, also know as AR(p), is used

where the model coefficients are denoted by α, the spanwise coordinate is y,

and l is the model lag.

**Coherence Analysis**

**Validation of spanwise AR model against DNS. ****DNS signals sampled at z+=15 in undisturbed flow. **

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**Schematic of MC strategy: **

**Matlab code (excerpts) implementing the MC algorithm**

**Coherence Analysis**

• MC code is used to determine the **significance limit c **of the coherence estimate:

where P is the probability operator, and α is the significance level (here α=99.9%).

• Note: A coherence value of 0.25 is “equivalent” to a correlation coefficient of

sqrt(0.25) = 0.5, which would indicate a strong statistical link between the signals.

• Physically, this would imply that the incoming boundary layer plays an important

role in driving the shock motion at low frequencies.

**Coherence Analysis**

Estimated coherence between the shock motion in the freestream and the

pressure signal at two locations in the upstream, undisturbed boundary layer.

**c=99.9% significance limit **

St = 0.063

**Dynamic Mode Decomposition (DMD) – Level 7 **

**(Rowley et al. 2009; Schmid 2010) **

• Consider a discrete linear system: x_{n+1 }= **A**x_{n }

• The eigenvalues and eigenvectors of matrix **A** can be found iteratively using the
classic Arnoldi algorithm.

• If the matrix **A** is not accessible, but a set of realizations {x_{0},x_{1},….,x_{n}} is, then an
alternative algorithm, called DMD, can be used.

• DMD can be applied to nonlinear systems (such as Navier-Stokes) since
knowledge of **A** is not required.

• DMD modes are ordered by their associated eigenvalues (frequencies).

DMD spectrum for DNS of shock wave/ turbulent boundary layer interaction, DMD analysis performed with Tu & Rowley (Princeton University).

• The DMD mode shows activity along the shock and in the separated flow.

• There is excellent qualitative agreement with the linear stability results available in the

literature. x/δ

z/

δ

St = 0.063: u, real part

**DMD and Link with Stability Theory **

Global linear stability theory (Pirozzoli et al. 2010, same mode as

Touber & Sandham 2009)

**u’ **_{ (}_{ρ}_{u)’ }

Low-frequency DMD mode from DNS

Conclusion: The unsteadiness in STBLI is driven by an instability of the separated flow, which is qualitatively captured by stability theory.

**CFD of Hypersonic STBLI**

• CFD of hypersonic STBLI requires robust, shock-capturing numerics that can

resolve shocks, transient shocklets, and compressible turbulence (WENO scheme well-suited).

• The highest Mach number reported in the literature to date is 3. There are no high-fidelity simulations (LES or DNS) in the hypersonic regime (M > 5), despite the practical relevance of STBLIs in this regime.

**Previous high-fidelity CFD simulations reported in the literature (representative selection) and their ****associated Mach numbers. **

**Authors ** **Methodology Configuration ** **Mach ** **Year **

Adams DNS Comp. ramp 3 2000

Loginov et al. LES Comp. ramp 2.95 2006

Morgan et al. LES Reflected shock 2.05 2010

Pirozzoli & Grasso DNS Reflected shock 2.25 2006

Priebe et al. DNS Reflected shock 2.9 2009

**Numerical Method & Computational Setup **

### •

**Inviscid fluxes:**

**Inviscid fluxes:**

### 4

th### -order linearly optimized Weighted Essentially

### Non-Oscillatory (WENO) method (Martin et al. JCP 06).

### •

**Viscous fluxes: **

**Viscous fluxes:**

### standard 4

th### -order central differences.

### •

**Time-integration: **

**Time-integration:**

### 3

rd### -order low-storage Runge-Kutta (Williamson JCP 80).

### •

### Massively-parallel DNS of Mach 7.2 STBLI:

– Total of 235 million grid points (for the auxiliary and principal DNS). – Run on 3200 cores of a Cray XE6.

– Total data set size 9 Terabytes.

– Total computational cost 3 million CPU hrs.

**Numerical Schlieren of auxiliary DNS ****(Priebe & Martin AIAA 2011) **

**Typical Instantaneous **

**Flow Field (Principal DNS) **

*Isosurface of magnitude of density gradient colored by z-coordinate (wall – blue; z=2δ _{ref} – white) *

• Broad validation effort: - Grid sensitivity study

- Comparison against known correlations - Comparison against experiments

• CFD flow conditions similar to those of experiments performed by Smits and co-workers at Princeton University (Bookey et al. 2005; Sahoo et al. 2009; Schreyer et al. 2011; Williams & Smits 2012).

**Code Validation for Hypersonic STBLI**

**Cross-validation with experiments at similar flow conditions possible ****PIV setup at the Gas Dynamics Lab, ****Princeton University (Photos courtesy ****O. Williams) **

**Code Validation I: Grid Convergence Study**

**Case ** **Nx ** **Ny ** **Nz ** **Lx/δ** **Ly/δ** **Lz/δ** **Inflow BC **

Baseline 1060 232 170 22.5 3.0 9.0 RRM

Refined in x 2120 232 170 22.5 3.0 9.0 RRM

Refined in y 1060 464 170 22.5 3.0 9.0 RRM

Refined in z 1060 232 340 22.5 3.0 9.0 RRM

Production 1060 768 170 22.5 10.0 9.0 Aux. DNS

**Code Validation II: Incoming Boundary Layer**

**Skin-friction coefficient in the incoming boundary layer: ****comparison of DNS against van Driest II theory. **

**Morkovin-scaled streamwise velocity **

**fluctuation u’ in the incoming boundary layer. ****Figure courtesy Williams & Smits, 2013. **

Comparison with known correlations Example:

Comparison with PIV experiments Example:

PIV, Williams & Smits (2013)

wall normal coordinate, z/δ

M orkovi n-s ca le d u’

Present DNS,

Instantaneous numerical Schlieren visualization

Experiment, Filtered Rayleigh Scattering Bookey, Wyckham, Smits & Martin (2005)

**Code Validation III: (Qualitative) Comparison **

**against Experiments **

Present DNS,

Mean numerical Schlieren visualization

### Experiment, Schlieren Schreyer et al. (2011)

*Shock angle:*

*~13° (exp),*

*~13° (DNS),*

*14.3° (inviscid theory).*

**Experimental work (PIV) is ongoing. Quantitative comparison will be made with the ****PIV results once they are available. **

• State-of-the-art RANS modeling for hypersonic STBLIs inaccurate. For example, Roy & Blottner (2006) conclude:

**“while some of the turbulence models do provide reasonable **

** predictions for the surface pressure, the predictions for surface heat **** flux are generally poor, and often in error by a factor of four or more.” **

• Uncertainty metric E_{CC} based on wall quantities (Gnoffo et al. 2011):

**Turbulence Models for Hypersonic STBLIs**

**Separation State ** **E _{CC}**

Attached 25%

Separated 55%

**RANS uncertainty for hypersonic ****compression ramp STBLIs ****(Gnoffo et al. 2011). **

**Comparison between heat transfer predicted by standard ****RANS models and the experiments by Murray (Brown 2011). **

**DNS dataset addresses above needs. **

**Quantities relevant to turbulence modeling (e.g. TKE and Reynolds-stress ****budgets, Reynolds-stress anisotropies, thermodynamic variables) may be ****extracted from DNS dataset to assess modeling assumptions. **

**Underlying causes for poor model performance: **

• Absence of high-fidelity (LES or DNS) simulations.

• Scarcity of high-quality experiments suitable for model development and validation. Settles & Dodson (1994) identify 7 suitable experiments; Roy & Blottner (2006) identify 9.

• Most validation datasets include wall quantities only.

• Limited knowledge of turbulence behavior in hypersonic STBLIs. • Experimental references investigating turbulence behavior:

*1. * *Mikulla & Horstman 1976 *
*2. * *Bookey et al. 2005 *

*3. * *Schrijer, Scarano & van Oudheusden 2006 *
*4. * *Schreyer, Sahoo & Smits 2011 *

*5. * *Williams & Smits 2012 *

• Possible modeling assumptions linking the velocity field to the temperature field:

• The accuracy of these assumptions may be verified using the DNS database.

**Turbulence Models for Hypersonic STBLIs**

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**pl**

**exi**

**exi**

**ty**

**ty**

• Assessment using DNS database:

**Turbulence Models for Hypersonic STBLIs**

### €

*Ra*

_{f}### =

### 2

*C*

*h*

*C*

_{f}### =

*const*

### ≈

### 1

**Reynolds analogy (RA), **

**links the coefficients of skin friction and heat transfer. **

• Heat transfer scales better with pressure than with skin friction. • Acceptable agreement with QP85 scaling (Back & Cuffel 1970).

**z/**b
**Pr**
**t**
0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
1.5
2
2.5
3

• Assessment using DNS database:

**Turbulence Models for Hypersonic STBLIs**

• Pr_{T} ≈ 1 in the bulk of the boundary layer downstream of the
compression corner.

• However, assumption Pr_{T} ≈ 1 breaks down near the wall.

### €

### Pr

_{T}### =

### ρ ʹ′

*u *

### ʹ′

*w *

### ʹ′

### ʹ′

### ρ ʹ′

*T *

### ʹ′

*w *

### ʹ′

### ʹ′

### ∂

*T *

### ˜

### ∂

*z*

### ∂

*u *

### ˜

### ∂

*z*

### =

*const*

### ≈

### 1

**Turbulent Prandtl number,**_{links the turbulent stress and heat transfer. }**x/****δ**** = 1.0 (downstream of ****compression corner) ****z/**b
**Pr**
**t**
0.6 0.8 1 1.2 1.4
-1
0
1
2
3
**x/****δ**** = 4.0 **

### •

### CFD code development effort – including both solver and post-processor

### development – to simulate and analyze STBLIs has been described.

### •

### Development of boundary conditions to ensure long-duration, uncorrelated

### inflow data.

### •

### Implementation and validation of LES capability.

### •

### Development of advanced post-processing tools, including spectral

### coherence estimation and associated error bounds.

### •

### Large-scale DNS of a hypersonic STBLI has been presented, including:

– Code validation

– Implications for developing and validating turbulence models

**Questions? **

**CFD of Hypersonic STBLI**

• Parameters that govern STBLI flows: Re, α, M, T_{w}/T_{r}, h.

• STBLIs are of practical importance in the hypersonic regime (Mach number > 5); they are relevant in reentry applications and scramjet propulsion, where they can, for example, lead to significant heat transfer damage.

**Thermal image of space shuttle on reentry, showing ****increased heat transfer due to roughness-induced ****transition (NASA) **

**parameters relevant for **

**high-speed and high-enthalpy flight **

**X-15 hypersonic vehicle (NASA) **

**X-15 fin heating damage (Watts 1968). **

**Upstream boundary layer **

Example from STBLI: Coherence between freestream shock motion and wall pressure.

• Shock unsteadiness statistically linked to the downstream

separated flow, weaker link with the upstream flow.

• Fluctuations near separation and reattachment out of phase.

**Shock motion region ****Reattachment region & downstream **

‘Pulsation’ of separated flow.

• Knowledge of error bounds enables precise inferences from

data. In this case, it shows that the correlation with the inflow boundary layer is weak.

• Coherence analysis is a powerful tool to find the statistical link

between different flow regions in a scale-decomposed manner.

ph

ase

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