Proper orthogonal decomposition of turbulent flow around a finite blunt plate

Abstract

In this article, proper orthogonal decomposition (POD) and Extended POD are applied to reveal the relationship between coherent vortices and wall physics of a flow over a finite blunt plate, with a Reynolds number ($Re_d$) of $1.58\times 10^4$. The flow is simulated by improved delayed detached eddy simulation method. The predicted results are in good agreement with previous experimental and numerical investigations in terms of statistical properties and unsteadiness. Four typical coherent structures are found by using POD, that is shedding mode, flapping mode, Kármán mode and force mode. Then, Extended POD is used to analyze the coherent structural features and their relation to wall pressure fluctuations. It is concluded that wall pressure fluctuations are a combined results of all of the main structural features with different significant influencing region, while lift unsteadiness is mainly attributed to wake dynamics, especially the force mode.

Graphical Abstract

Appendices

Appendix 1: Governing equations

The incompressible RANS equations in the Cartesian coordinate system read

$$\begin{aligned} \left. \begin{aligned} \frac{\partial {u_i}}{\partial {t}} + u_j \frac{\partial {u_i}}{\partial {x_j}}&= -\frac{1}{\rho }\frac{\partial {p}}{\partial {x_i}} + (\nu _l + \nu _{t})\frac{\partial ^2{u_i}}{\partial {x_i}\partial {x_j}} \\ \frac{\partial {u_i}}{\partial {x_i}}&= 0 \end{aligned} \right\} , \end{aligned}$$

(2)

where $u_i$ is the averaged velocity component in $\varvec{e}_i$, $x_i$ is the coordinate in $\varvec{e}_i$, p is averaged pressure, $\rho$ is the fluid density, $\nu _l$ and $\nu _{t}$ are the molecular kinematic viscosity and turbulent kinematic viscosity (eddy viscosity), respectively. $\nu _{t}$ is given by turbulence models.

For the purpose of obtaining high-fidelity numerical results at high Reynolds numbers for wall-bounded turbulent flow under the condition of limited computing resources, Spalart et al. (1997) proposed the detached eddy simulation (DES) method, which combined the strengths of LES and RANS, based on the S-A turbulence model (Spalart and Allmaras 1992). DES switches to the LES branch in the regions faraway from the wall, and to the RANS branch in the regions near the wall. Inspired by Spalart et al. (1997), Strelets (2001) further developed the DES method based on the Menter $k{-}w{-}$SST model (Menter 1994; Menter et al. 2003). The SST model equations read

$$\begin{aligned} \left. \begin{aligned} \frac{\partial {k}}{\partial {t}}+u_j\frac{\partial {k}}{\partial {x_j}}&= P_k + {\mathrm{{{Diff}}}}_k - D_k \\ \frac{\partial {\omega }}{\partial {t}}+u_j\frac{\partial {\omega }}{\partial {x_j}}&= P_{\omega } + {\mathrm{{{Diff}}}}_{\omega } - D_{\omega } + \left( 1 - F_1\right) {CD}_{k\omega } \end{aligned} \right\} , \end{aligned}$$

(3)

where $P_k, {\mathrm{{{Diff}}}}_k, D_k$ are the production, diffusion and dissipation terms in the equation for k (turbulent kinetic energy), respectively, while $P_{\omega }, {\mathrm{{{Diff}}}}_{\omega }, D_{\omega }, {CD}_{k{\omega }}$ are the production, diffusion, dissipation and cross-diffusion terms in the equation for w (dissipation rate of k), respectively. $F_1$ is a blending function. The eddy viscosity is calculated with

$$\nu _t = \min \left( \frac{k}{\omega }, \frac{a_1 k}{S F_2} \right) .$$

(4)

where S is the strain rate $S=\frac{1}{2}\left( \frac{\partial {u_i}}{\partial {x_j}} + \frac{\partial {u_j}}{\partial {x_i}}\right)$, $F_2$ is a blending function and $a_1$ is a constant. Details of the terms were given in the Ph.D. thesis of Hu (Hu 2016).

Strelets (2001) extended the original DES method based on Menter $k{-}w{-}$SST model by replacing the dissipation term,

$$D_k = \beta ^* k\omega = k^{3/2}/l_{\mathrm{{RANS}}},$$

(5)

in the k equation with,

$$D_k = \beta ^* k\omega = k^{3/2}/l_{\mathrm{{DES}}},$$

(6)

where $l_{\mathrm{{RANS}}} = k^{1/2} / \left( {\beta }^* \omega \right)$ is the length scale of the $k-w-$SST model, $\beta ^*=0.09$, $l_{\mathrm{{DES}}} = \min \left( l_{\mathrm{{RANS}}},l_{\mathrm{{LES}}} \right)$ is the DES length scale, $l_{\mathrm{{LES}}}=C_{\mathrm{{DES}}}\Delta$ is the LES length scale and $\Delta$ is the grid scale.

The DES methods were found to have the “grid-induced separation” and the “log-layer mismatch” problems. They were tackled by introducing a delay function by Spalart et al. (Spalart et al. 2006) and redefining the grid scale by Shur et al. (Shur et al. 2008), respectively. Shur et al. (Shur et al. 2008) proposed the length scale of IDDES,

$$l_{\mathrm{{{IDDES}}}} = {\tilde{f}}_d \left( 1 + f_e\right) l_{\mathrm{{RANS}}} + \left( 1 - {\tilde{f}}_d \right) l_{\mathrm{{LES}}},$$

(7)

where ${\tilde{f}}_d$ is a blending function and $f_e$ is an “elevation function.”

And the new definition of the grid scale is

$$\Delta = \min \{ \max [ C_{\mathrm{{w}}} d_{\mathrm{{w}}}, C_{\mathrm{{w}}} h_{\mathrm{{max}}}, h_{\mathrm{{wn}}} ], h_{\mathrm{{max}}} \},$$

(8)

where $d_{\mathrm{{w}}}$ is the wall-normal distance, $h_{\mathrm{{max}}}$ is the maximum local grid spacing, $h_{\mathrm{{wn}}}$ is the wall-normal local grid spacing, the constant $C_{\mathrm{{w}}}=0.15$. This definition depends on not only the grid size, but also the wall-normal distance.

Readers are referred to Menter et al. (2003), Spalart et al. (2006) and Shur et al. (2008) for more details.

Appendix 2: Extended-POD technique

The POD technique, which was first introduced in fluid dynamics by Lumley (1967), decomposes instantaneous flow variable ($\Phi (t,{\mathbf {x}})$ can be a scalar or vector) into averaged quantity and fluctuation. Then, the fluctuation is represented by a linear combination of mutually orthogonal basis functions. The decomposition is

$$\Phi (t,{\mathbf {x}}) = {\overline{\Phi }}({\mathbf {x}}) + \Phi ^\prime (t,{\mathbf {x}}) = {\overline{\Phi }}({\mathbf {x}}) + \sum ^{\infty }_{i=1} a_{\Phi }^i(t) \phi _{\Phi }^i({\mathbf {x}}),$$

(9)

where t is time, ${\mathbf {x}}\in \Omega$ and $\Omega$ is the spatial domain on which the POD technique will be applied. ${\overline{\Phi }}$ and ${\Phi }^\prime$ stand for time-average and fluctuation of $\Phi ,$ respectively. The basis functions $\phi _{\Phi }^i({\mathbf {x}})$, known as POD modes of variable $\Phi$, represent the flow pattern. The coefficient, $a_{\Phi }^i(t)$, is responsible for the evolution in time.

The snapshot POD algorithm (Sirovich 1987; Meyer et al. 2007) was used in this paper, in consideration of the larger dimensionality in space than in time. The snapshot matrix is $X = [\Phi ^{\prime }\left( t_1,{\mathbf {x}} \right) ,\ldots ,\Phi ^{\prime }\left( t_m,{\mathbf {x}} \right) ]$, where $\Phi ^{\prime }\left( t_j,{\mathbf {x}} \right) (j = 1,\ldots ,m)$ are fluctuation fields with a spatial dimensionality of n. By solving the eigenvalue problem

$$X^* X y_i = \lambda _i y_i, \quad i = 1,\ldots ,m,$$

(10)

the POD modes and coefficients are obtained,

$$\begin{aligned}\phi _{\Phi }^i({\mathbf {x}})&= X y_i /\left\| X y_i \right\| _{F}, \\ a_{\Phi }^i \left( t_j\right)&= (\phi _{\Phi }^i({\mathbf {x}}))^* \Phi ^{\prime }\left( t_j,({\mathbf {x}})\right) , \end{aligned}$$

(11)

where $(\,)^*$ stands for the complex-conjugate transpose, $\left\| \right\| _{F}$ is the Frobenius norm. In Eq. (10), the eigenvalues $\lambda _i$ stand for the energies associated with the POD modes $\phi _{\Phi }^i$. Hence, the modes are rearranged in descending order in accordance with $\lambda _i$.

Borée (2003) provided a general definition of EPOD mode:

$$\psi _{\Psi }^{i}({\mathbf {y}}) = \frac{1}{m}\sum ^{m}_{j=1} \frac{a_{\Phi }^i(t_j) \Psi ^\prime (t_j,\quad {\mathbf {y}})}{\lambda _i},\quad {\mathbf {y}} \in \Omega ^\dag ,$$

(12)

where $\Psi$ is a flow variable ($\Psi$ can be the same with or different from $\Phi$) on the domain $\Omega ^\dag$ ($\Omega ^\dag$ can be equal to $\Omega$, can contain $\Omega$ or not). Each realization of $\Psi$ is associated with a realization of $\Phi$ on $\Omega$. For each realization of $\Psi$, the following decomposition is proposed:

$$\begin{aligned} \Psi ^{\prime }_c(t,{\mathbf {y}})&= \sum ^{m}_{i=1} a_{\Phi }^i(t) \psi _{\Psi }^{i}({\mathbf {y}}), \\ \Psi _d(t,{\mathbf {y}})&= \Psi (t,{\mathbf {y}}) - \Psi ^{\prime }_c(t,{\mathbf {y}}), \end{aligned}$$

(13)

where $\Psi ^{\prime }_c(t,{\mathbf {y}})$ and $\Psi _d(t,{\mathbf {y}})$ are correlated part (in terms of $\Phi$) and uncorrelated part, respectively.

The EPOD mode $\psi _{\Psi }^{i}$ are the correlated part in the domain $\Omega ^\dag$, associated with the physical process represented by the POD mode $\phi _{\Phi }^i$ in the domain $\Omega$. Hence, this technique can be used, for example, to study the spatial and temporal interactions between coherent structures of a flow (Hoarau et al. 2006).