Coherent pressure and acceleration estimation from triply decomposed turbulent bluff-body wakes

Abstract

A novel method is introduced to estimate phase-averaged pressure and acceleration of the coherent structures. It is applied and validated in the near wake of a two-dimensional bluff body. The method is validated using synthetic data generated by simulating an unsteady laminar flow around a D-shaped bluff body; the effect of noise is investigated to simulate an experimental scenario. The acceleration terms, required for the closure of flow-governing equations, are accurately estimated using this approach. The findings demonstrate that the methodology outputs consistent results for the pressure field while combined with a divergence-correction scheme, based on Helmholtz decomposition, and an optimal control method using sparse sensor measurements. The approach is later validated on experimental data obtained in the turbulent wake of a bluff body by means of “Particle Image Velocimetry” (PIV). Triple-decomposed flow fields are provided as input, where triple-decomposition refers to the instantaneous flow fields split into a mean flow, coherent and incoherent structures. The coherent structures are extracted using the method of “Proper Orthogonal Decomposition” (POD). Coherent lift coefficients, estimated using integral budgets of momentum, remain within the sensor-measured range, whereas coherent drag coefficients show excellent agreement. Moreover, a significant reduction in the residuals, arising from the balance of the flow-governing equations, is achieved. Although the temporal information is lost when computing POD modes, the proposed method is able to recover it and estimate the pressure. Accordingly, the method can be extended to any time-dependant modal decomposition, thereby providing a low-cost alternative to expensive “time-resolved” PIV (tr-PIV) measurements.

Appendix

1.1 2D Divergence correction

The divergence-free correction is based on the Helmholtz decomposition, which for a velocity vector reads:

$$\begin{aligned} {\tilde{\textbf{u}}}= \nabla {\tilde{\varphi }} + \nabla \times {\tilde{\mathbf {\psi }}}. \end{aligned}$$

(22)

The vector potential ${\tilde{\mathbf {\psi }}}$ is divergence-free by construction. For a two-dimensional incompressible rotational flow, the scalar potential part vanishes and the vector potential becomes ${\tilde{\mathbf {\psi }}}=(0,0,{\tilde{\psi }}_{z})$. Hence, the velocity components are given by $({\tilde{u}},{\tilde{v}},0)=\left( {\partial {\tilde{\psi }}_{z}}/{\partial y},-{\partial {\tilde{\psi }}_{z}}/{\partial x},0\right) ,$ where the vector potential becomes the stream function. Equation (22) can be rewritten as $\nabla \times {\tilde{\textbf{u}}}=-\varDelta {\tilde{\psi }}_{z}.$ The corrected velocity vector is obtained by solving the Poisson equation for stream function with Neumann boundary conditions.

1.2 Optimal control

The method is an optimisation problem with an objective to match the estimated pressure to the sparsely measured pressure, for example: using pressure sensors (Shanmughan et al. 2020). Then, the cost function takes the form

$$\begin{aligned}{} & {} {\mathcal {J}}\left( {\tilde{p}},\varphi \right) =\frac{1}{2}\int _\varGamma H\left( {\tilde{p}}_c-{\tilde{p}}_{ref}\right) ^2 d\varGamma \nonumber \\{} & {} \quad + \frac{\gamma }{2}\int _\varGamma \varphi ^2 d\varGamma , \end{aligned}$$

(23)

where $\varphi$ and the subscript c refer to the control applied on the boundary and the computed values, respectively. When considering phase-averaged pressure, the state constraints read,

$$\begin{aligned}&\varDelta {\tilde{p}} = f_{1}\left( U_{i}, {\tilde{u}}_{i}, {u}^{{{\,\mathrm{\prime \prime }\,}}}\right) \, \text{ in } \, \varOmega \text {,} \quad \nabla {\tilde{p}} \cdot n \nonumber \\&\quad = g_{1}\left( U_{i}, \partial _{t}u_{i}, {\tilde{u}}_{i}, {u}^{{{\,\mathrm{\prime \prime }\,}}}\right) +\varphi \, \text{ on } \, \varGamma , \end{aligned}$$

(24)

which are the Poisson equation and Neumann boundary conditions respectively for the phase-averaged pressure. Then, the Lagrangian function, using the formal Lagrange multiplier method (Tröltzsch 2010), may be formulated as,

$$\begin{aligned}{} & {} {\mathcal {L}}({\tilde{p}},\varphi ,\lambda ^+,\varphi ^+)={\mathcal {J}}({\tilde{p}},\varphi )-\langle \varDelta {\tilde{p}} - f_{1},\lambda ^+ \nonumber \\{} & {} \quad \rangle _\varOmega - \langle \nabla {\tilde{p}} \cdot n - g_{1}-\varphi , \varphi ^+ \rangle _\varGamma \end{aligned}$$

(25)

The objective then is to minimise ${\mathcal {L}} (P,\varphi ,\lambda ^+,\varphi ^+)$, i.e. ${\mathcal {D}}_{{\tilde{p}}}{\mathcal {L}}=0$ and ${\mathcal {D}}_{\varphi }{\mathcal {L}}=0$, where ${\mathcal {D}}$ represents the Fréchet derivative. From the former, the adjoint state equations can be derived as,

$$\begin{aligned}&\varDelta \lambda ^+=0 \, \text{ in } \, \varOmega ,\, \, \nabla _n\lambda ^+\nonumber \\&\quad {{\varvec{ \quad }}}=H({\tilde{p}}_c-{\tilde{p}}_{ref}) \, \text{ on } \, \varGamma , \, \, \lambda ^+=-\varphi ^{+} \, \text{ on } \, \varGamma , \end{aligned}$$

(26)

and from the latter, the optimal solution for the control is found as

$$\begin{aligned} \varphi ={\lambda ^+}/{\gamma } \, \text {in} \, \varGamma . \end{aligned}$$

(27)

A first guess of the constant arising out of the inversion of the Poisson equation is computed as the average difference between the estimated and the true pressure. It is then subtracted from the estimated pressure prior to the application of control and updated iteratively. The Neumann boundary conditions are corrected until the estimate converges to the true pressure.

Algorithm 1

Two-dimensional reconstruction of coherent pressure and acceleration fields.