Multi-scale study of the transitional shock-wave boundary layer interaction in hypersonic flow

Abstract

A high-fidelity simulation of the massively separated shock/transitional boundary layer interaction caused by a 15-degrees axisymmetrical compression ramp is performed at a free stream Mach number of 6 and a transitional Reynolds number. The chosen configuration yields a strongly multiscale dynamics of the flow as the separated region oscillates at low-frequency, and high-frequency transitional instabilities are triggered by the injection of a generic noise at the inlet of the simulation. The simulation is post-processed using Proper Orthogonal Decomposition to extract the large scale low-frequency dynamics of the recirculation region. The bubble dynamics from the simulation is then compared to the results of a global linear stability analysis about the mean flow. A critical interpretation of the eigenspectrum of the linearized Navier–Stokes operator is presented. The recirculation region dynamics is found to be dominated by two coexisting modes, a quasi-steady one that expresses itself mainly in the reattachment region and that is caused by the interaction of two self-sustained instabilities, and an unsteady one linked with the separation shock-wave and the mixing layer. The unsteady mode is driven by a feedback loop in the recirculation region, which may also be relevant for other unsteady shock-motion already documented for shock-wave/turbulent boundary layer interaction. The impact of the large-scale dynamics on the transitional one is then assessed through the numerical filtering of those low wavenumber modes; they are found to have no impact on the transitional dynamics.

Appendices

Appendix 1: Noise injection

This section is dedicated to the technical description of the white noise injection in the QDNSs following the procedure described in [6]. The injection is realized 4 cells downstream ($i=4$) of the inlet boundary condition in order not to interfere with it. The form of this injection is the following :

$$\begin{aligned} \rho '[j,k]=\rho [j,k] (1+ 0.01 r_{n}[j,k]) \end{aligned}$$

(3)

With $r_{n}$ a random number normalized such that the root-mean-square on the whole injection plane is 1:

$$\begin{aligned} r_{n}[j,k] = \frac{r_{r}[j,k]}{\sqrt{\overline{r_{r}^2}}} \end{aligned}$$

(4)

$r_{r}$ being a random number from a continuous uniform distribution between $-0.5$ and 0.5, $\overline{.}$ being for this special case a spatial average and j, k ranging the indices of the cell of the injection plan (i.e. $j\in [0,60]$ for the wall normal direction and $k\in [0,k_{max}]$ in the azimuthal direction). As the time step used in the computation is far less than the convection time through one cell in the streamwise direction, the spatial scheme would be unable to transport a white noise that is updated every iteration. To address this issue, it was chosen not to update the noise every iteration, but to keep it constant for 15 iterations between each update. This ensures that the scheme can discretize the noise while the spectral content is still rich enough in the high frequencies for the present study, a power spectral density of pressure perturbations created by the noise is presented in Fig. 2.

Appendix 2: Azimuthal decomposition of the Jacobian operator

Since the solver FAST works internally with Cartesian coordinates, one has to first carry out a transformation to cylindrical coordinates to retrieve the axisymmetry of the flow using the following relation

$$\begin{aligned} \left( \begin{array}{c} \rho \\ \rho {u_x}\\ \rho {u_r}\\ \rho u_\theta \\ \rho E \\ \end{array} \right) = \left[ \begin{array}{ccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad cos(\theta ) &{}\quad sin(\theta ) &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -sin(\theta ) &{}\quad cos(\theta ) &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ \end{array} \right] \left( \begin{array}{c} \rho \\ \rho {u_x}\\ \rho {u_y}\\ \rho u_z \\ \rho E \\ \end{array} \right) \end{aligned}$$

(5)

Under appropriate indexing of the degrees of freedom, the Jacobian operator can then be rearranged into the block-circulant form

$$\begin{aligned} {\mathbf {J}} = \left[ \begin{array}{ccccc} {\mathbf {A}}_0 &{}\quad {\mathbf {A}}_1 &{}\quad ... &{}\quad {\mathbf {A}}_{n-2} &{}\quad {\mathbf {A}}_{n-1}\\ {\mathbf {A}}_{n-1} &{}\quad {\mathbf {A}}_0 &{}\quad ... &{}\quad {\mathbf {A}}_{n-3}&{}\quad {\mathbf {A}}_{n-2}\\ {\mathbf {A}}_{n-2} &{} \quad {\mathbf {A}}_{n-1} &{}\quad ... &{}\quad {\mathbf {A}}_{n-4} &{}\quad {\mathbf {A}}_{n-3}\\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots \\ {\mathbf {A}}_1 &{}\quad {\mathbf {A}}_2&{}\quad ... &{}\quad {\mathbf {A}}_{n-1} &{}\quad {\mathbf {A}}_{0}\\ \end{array} \right] , \end{aligned}$$

(6)

where each line of blocks corresponds to a given azimuthal slice of the mesh and the block matrices ${\mathbf {A}}_0$, ..., ${\mathbf {A}}_{n-1}$ have a size corresponding to such a slice (the size of a 2D problem). The block-circulant nature of the matrix comes from the numerical and physical equivalence of all azimuthal slices of the mean flow, which cannot be distinguished from one another. As shown by Schmid et al. [40], this block circulant matrix can then be transformed into a block-diagonal matrix

$$\begin{aligned} {\tilde{\mathbf{J}}} = \left[ \begin{array}{ccccc} {\tilde{\mathbf{A}}}_0 &{} &{} &{} &{} \\ &{} {\tilde{\mathbf{A}}}_1 &{} &{} &{}\\ &{} &{} \ddots &{} &{}\ \\ &{} &{} &{} &{} {\tilde{\mathbf{A}}}_{n-1}\\ \end{array} \right] , \end{aligned}$$

(7)

with

$$\begin{aligned} {\tilde{\mathbf{A}}}_m = {\mathbf {A}}_0 + \rho _m {\mathbf {A}}_1 + \rho _m^2 {\mathbf {A}}_2 + ... + \rho _m^{n-1} {\mathbf {A}}_{n-1}, \end{aligned}$$

(8)

and $\rho _m=e^{\frac{j 2 \pi m}{n}}$ corresponding to a m-root of unity.

This leads to independent studies for each wavenumber of interest through the study of the eigenvalues of the reduced operator ${\tilde{\mathbf{A}}}_m$ instead of the full operator ${\mathbf {J}}$.

Appendix 3: Numerical procedure for the proper orthogonal decomposition

First, $N_{r}=600$ snapshots are randomly sampled from the pool of filtered snapshots, then a Discrete Fourier Transform (DFT) is applied in the azimuthal direction, giving Fourier mode vectors ${\hat{\mathbf{S}}}^{k}\!(m)$, where k is the realisation number and m the azimuthal wavenumber of the mode. Due to the spectral transformation in the azimuthal direction, the vectors ${\hat{\mathbf{S}}}^{k}\!(m)$ correspond to bi-dimensional fields: they contain complex values associated with each flow variables at each pair (x, r) from the mesh. For a given wavenumber (m) of interest, the Fourier modes of all realisations are then stacked in a matrix $\hat{{\mathbf {X}}}_{m}$, which reads

$$\begin{aligned} {\hat{\mathbf{X}}}_{m} =\left[ {\hat{\mathbf{S}}}^{0}\!(m), \ {\hat{\mathbf{S}}}^{1}\!(m), \ \cdots , \ {\hat{\mathbf{S}}}^{N_r-1}\!(m) \right] . \end{aligned}$$

(9)

This matrix is then processed similarly to a snapshot matrix in a classical POD decomposition: the i-th POD mode $\mathbf {\phi _i}^{(m)}$ can be computed from the i-th left singular vector of ${\hat{\mathbf{X}}}_{m}$, which may be computed by solving the eigenproblem associated with the cross spectral density matrix

$$\begin{aligned} {\hat{\mathbf{X}}}_{m} {\hat{\mathbf{X}}}_{m}^\star \mathbf {Q_e} \ \mathbf {\psi _i}^{(m)} = \lambda _i \mathbf {\psi _i}^{(m)}, \end{aligned}$$

(10)

with $\mathbf {Q_e}$ the inner product associated with the energy norm defined by Chu [50] :

$$\begin{aligned} {{\mathbf{Q}}_{\mathbf{e}}}= & {} \Omega \left[ \begin{array}{ccccc} \frac{|\mathbf {\overline{u}}|^2 + R \overline{T}}{\overline{\rho }} + a_1 a_2^2 &{}\quad \frac{-\overline{u_x} (1+a_1 a_2)}{\overline{\rho }} &{}\quad \frac{-\overline{u_r} (1+a_1 a_2)}{\overline{\rho }} &{}\quad \frac{-\overline{u_\theta } (1+a_1 a_2)}{\overline{\rho }} &{}\quad \frac{a_1 a_2}{\overline{\rho }} \\ \frac{-\overline{u_x} (1+a_1 a_2)}{\overline{\rho }} &{}\quad \frac{\rho +\overline{u_x^2}a_1}{\overline{\rho ^2}} &{}\quad \frac{\overline{u_x u_r}a_1}{\overline{\rho ^2}} &{}\quad \frac{\overline{u_x u_\theta }a_1}{\overline{\rho ^2}} &{}\quad - \frac{\overline{u_x}a_1}{\overline{\rho ^2}} \\ \frac{-\overline{u_r} (1+a_1 a_2)}{\overline{\rho }} &{}\quad \frac{\overline{u_x u_r}a_1}{\overline{\rho ^2}} &{}\quad \frac{\rho +\overline{u_r^2}a_1}{\overline{\rho ^2}} &{}\quad \frac{\overline{u_r u_\theta }a_1}{\overline{\rho ^2}} &{}\quad - \frac{\overline{u_r}a_1}{\overline{\rho ^2}} \\ \frac{-\overline{u_\theta } (1+a_1 a_2)}{\overline{\rho }} &{}\quad \frac{\overline{u_x u_\theta }a_1}{\overline{\rho ^2}} &{}\quad \frac{\overline{u_r u_\theta }a_1}{\overline{\rho ^2}} &{}\quad \frac{\rho +\overline{u_\theta ^2}a_1}{\overline{\rho ^2}} &{}\quad -\frac{\overline{u_\theta }a_1}{\overline{\rho ^2}} \\ \frac{a_1 a_2}{\overline{\rho }} &{}\quad -\frac{\overline{u_x} a_1}{\overline{\rho ^2}} &{}\quad -\frac{\overline{u_r} a_1}{\overline{\rho ^2}} &{}\quad -\frac{\overline{u_\theta } a_1}{\overline{\rho ^2}} &{}\quad \frac{a_1}{\overline{\rho ^2}} \\ \end{array} \right] \end{aligned}$$

(11)

$$\begin{aligned} a_1= & {} \frac{\overline{\rho }}{C_v \overline{T}} \end{aligned}$$

(12)

$$\begin{aligned} a_2= & {} \frac{\frac{|\overline{{\mathbf {u}}}|^2}{2} - \overline{e}}{\overline{\rho }} \end{aligned}$$

(13)

With $\Omega$ the local cell volume and e the internal energy. This norm is commonly used in order to describe fluctuation energy in compressible flow [51,52,53] and is more adapted than a simple kinetic energy norm often used for (quasi-)incompressible flows. The POD modes are ordered with respect to their contribution to the global dynamics, i.e. $\lambda _0> \lambda _1> \lambda _2>\dots$, and for a given wavenumber (m), the relative contribution of the i-th POD mode is measured by the ratio $r_i = \lambda _i / \sum _k \lambda _k$.

In practice, the eigenmodes are computed by using the snapshots method of [54] which is a less costly but equivalent decomposition based on $\hat{{\mathbf {X}}}_{m}^\star \mathbf {Q_e} \hat{{\mathbf {X}}}_{m}$ rather than (10). This provides the right singular vectors of $\hat{{\mathbf {X}}}_{m}$, from which one can easily retrieve the POD modes (see for instance [32] for details). Note that it is possible to localise the POD analysis to a given region of the flow by setting all coefficients of $\mathbf {Q_e}$ associated with cells outside of this region to zero. The eigenvalue is then defined as the energy restrained to this specific zone.