Inflow and initial conditions for direct numerical simulation based on adjoint data assimilation
Context
In order to limit the computational cost associated to direct numerical (DNS) or large-eddy simulations (LES) of spatially evolving flows, new strategies to simulate only the region of interest of the flow have recently been developed [1]. A potential implication is that any results computed may be strongly influenced by the prescribed instantaneous inflow velocity profiles. These profiles are practically never available, and a usual practice is to generate synthetic inflow data satisfying certain properties, which may be known from experimental data or empirical correlations. However, the proposed stategies are divided into two separate parts. In a first step, the inflow condition is built from the data and then the simulation is conducted from these information. If better simulations are possible, something useful may be learnt from optimal control theory.
Description
This study presents a new approach based on variational data assimilation (VDA), generating simultaneous transitional initial and inflow boundary conditions and reproducing the spatiotemporal dynamics of the targeted flow. VDA is a technique derived from optimal control theory. It is expressed as the minimization with respect to a control variable of an ob jective function that measures a discrepancy between a state variable and noisy measurements, sub ject to a constraint given by the state variable dynamics. The control variable may be for instance a parameter of the dynamics or the initial condition [2]. Assuming that both the model and the objective function are differentiable, VDA proposes to solve this inverse problem looking for a control that cancels out the gradient of this cost function through the use of adjoint minimization techniques. These techniques enable to compute the functional gradient by means of the adjoint of the tangent linear dynamics. The tangent linear dynamics and its adjoint are provided by TAPENADE an automatic differentiation (AD) tool [4].
The dynamical model that relates the state function to the unknown of our inverse problem is based on pressure-velocity formulation of Navier-Stokes equations and is resolved by DNS. Coupled with this dynamics we consider available noisy measurements of the velocity at discrete instants separated by a given latency (much larger than the DNS time step). Tangent (forward mode AD) and adjoint (reverse mode AD) codes were generated for this DNS code. We performed validation experiments to check for correctness of both AD codes.
The minimization problem reads
with 
the control variables.
To minimize the objective function, the gradient functional is obtained by a forward integration of the dynamical system
followed by a backward integration of an adjoint dynamical model given by
Results
Control on the Initial Condition
To evaluate the proposed technique, we carried out a VDA experiment applied to the identification of the initial condition from a spatialy evolving 2D mixing layer flow by using numerical data. In this case the functional J depends only on the initial condition, and comes to an initial value control problem. Figure 1 indicates that the true state at the end of the assimilation window (a) is recovered with a good approximation (d) from the perturbed one (b), showing the quality of the derivatives obtained.
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Control of the Initial and Inflow condition
In order to assess the performance of the VDA method for the specification of initial and inflow conditions we have constituted a benchmark composed of DNS results and experimental PIV data. The numerical simulation and the experimental data concern both a wak behind a circular cylinder at Reynolds 125 and 200 respectively.
Figures below show obtained results. When the inlet of the assimilation domain is located downstream of the vortex formation region (VDA test A), initial guess stay close to the observation data and the optimization algorithm requires 80 iterations to reach the assimilated trajectory. On the other hand, if the inlet belongs to the vortex formation region (VDA test B), Taylor's hypothesis does not hold and the initial approximation is far from the data. this leads to an increase of the number of iterations required to get the assimilated solution.
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| Isocontour of the vorticity in the MLA | Isocontour of the vorticity in the MLB |
Results indicate that despite low spatial and temporal resolution observations, assimilated state exhibits fine scale details of vorticity, like vortex filaments. Furthermore, it should be noted that the proposed method provides a means to simulate a wake flow without simulating the flow around the obstacle.
References
A. Gronskis, D. Heitz and E. Mémin. Inflow and Initial conditions for direct numerical simulation based on adjoint data assimilation. In TSFP7, Ottawa, July 2011.
