This work extends our previous study from S. Shrestha et al. (2024) by introducing a new abstract framework for Variational Multiscale (VMS) methods at the discrete level. We introduce the concept of what we define as the optimal projector and present a discretisation approach that yields a numerical solution closely approximating the optimal projection of the infinite-dimensional continuous solution. In this approach, the infinite-dimensional unresolved scales are approximated in a finite-dimensional subspace using the numerically computed Fine-Scale Greens’ function of the underlying symmetric problem. The proposed approach involves solving the VMS problem on two separate meshes: a coarse mesh for the full PDE and a fine mesh for the symmetric part of the continuous differential operator. We consider the 1D and 2D steady advection–diffusion problems in both direct and mixed formulations as the test cases in this paper. We first present an error analysis of the proposed approach and show that the projected solution is achieved as the approximate Greens’ function converges to the exact one. Subsequently, we demonstrate the working of this method where we show that it can exponentially converge to the chosen optimal projection. We note that the implementation of the present work employs the Mimetic Spectral Element Method (MSEM), although, it may be applied to other Finite/Spectral Element or Isogeometric frameworks. Furthermore, we propose that VMS should not be viewed as a stabilisation technique; instead, the base scheme should be inherently stable, with VMS enhancing the solution quality by supplementing the base scheme.

This study is a collaborative effort within the NATO Science & Technology Organization, bringing together multiple institutions to advance reduced-order modeling. Aerodynamic reduced-order models were developed using two pseudorandom binary sequence (PRBS) training maneuvers, where the angle of attack and pitch rate varied in a periodic, deterministic manner with white-noise-like properties. The first maneuver maintained a constant Mach number of 0.85, while the second varied Mach from 0.1 to 0.9. The test case involved a generic triple-delta wing, simulated using the DoD HPCMP CREATE™-AV/Kestrel/Kestrel tools. Prescribed-body motion was used to vary input parameters under given freestream conditions. The resulting models predicted static and stability derivatives across different angles of attack and Mach numbers. They were also used to predict aerodynamic responses to arbitrary motions, including sinusoidal, chirp, Schroeder, and step inputs, showing good agreement with full-order data. Additionally, models predicting surface pressure accurately captured upper surface pressures across different spanwise and chordwise locations for both static and dynamic conditions.

The authors present an energy-biased training approach for predicting exact closure terms of the Navier-Stokes equations discretised in a Variational Multiscale framework. The approach initially involves formulating a constrained objective function, which is transformed into an unconstrained problem suitable for neural network training using the Augmented Lagrangian method. The constraints induce a prediction bias by enforcing the predicted closure term energy contributions to be smaller than the exact values, both for excessive backscatter or insufficient dissipation. Effectively, the approach controls energy contributions through a-priori measures rather than a-posteriori measures. The approach was applied to obtain the H¹ projection of a Re_τ = 180 turbulent channel flow on a 32 × 32 × 32 uniform mesh. Eight MLPs were trained to predict the closure terms associated with the weighting functions of each linear hexahedral element. Each network used 217 features, and five hidden layers with 600 neurons each. The final data-to-parameter ratio was ∼19.6: 1 (30 932 992: 1 575 604) per neural network. A-priori evaluation of the networks’ outputs demonstrated its ability to predict closure terms yielding the desired behaviour in energy transfer. This was true for both closure terms that yield energy gain and energy loss. In spite of the energy bias, the closure term predictions retained correlations greater than 0.85 with their exact value for all positions between the channel walls.

In this paper, we build on the work of Hughes and Sangalli (2007) dealing with the explicit computation of the Fine-Scale Greens’ function. The original approach chooses a set of functionals associated with a projector to compute the Fine-Scale Greens’ function. The construction of these functionals, however, does not generalise to arbitrary projections, higher dimensions, or Spectral Element methods. We propose to generalise the construction of the required functionals by using dual functions. These dual functions can be directly derived from the chosen projector and are explicitly computable. We show how to find the dual functions for both the L² and the H₀¹ projections. We then go on to demonstrate that the Fine-Scale Greens’ functions constructed with the dual basis functions consistently reproduce the unresolved scales removed by the projector. The methodology is tested using one-dimensional Poisson and advection–diffusion problems, as well as a two-dimensional Poisson problem. We present the computed components of the Fine-Scale Greens’ function, and the Fine-Scale Greens’ function itself. These results show that the method works for arbitrary projections, in arbitrary dimensions. Moreover, the methodology can be applied to any Finite/Spectral Element or Isogeometric framework.

Efficient input data generation for reduced-order model applications to accurately predict aerodynamic performance and stability characteristics over a large part of a fighter aircraft’s flight envelope is a major challenge. In this paper, aerodynamic reduced-order models are created from two pseudorandom binary sequence (PRBS) training maneuvers. During these maneuvers, the angle of attack and pitch rate change in a periodic and deterministic manner which is characterized by white-noise-like properties. Typical PRBS signals include sudden input variations between two distinct values, such as minimum and maximum angles of attack. However, the signals used in this paper were modified to have the step changes to depend on the simulation time. In the first motion, the aircraft undergoes a signal at a constant Mach number of 0.85. In the second motion, the Mach number varies in an optimized manner from 0.1 to 0.9. The test case is a generic triple-delta wing configuration. Simulations were run using the DoD HPCMP CREATE^RM-AV/Kestrel simulation tools. A prescribed-body motion was used to vary input parameters under given freestream conditions (Mach number and angle of attack). Different reduced-order methods were applied, that comprise regression, feed-forward neural network and auto-regressive surrogate modeling techniques to predict integrated force and moment coefficients and a proper-orthogonal decomposition based neural network approach for surface pressure prediction. Once models of integrated forces and moments were created, they were used to predict static and stability derivatives at different angles of attack and Mach numbers. Models were then used to predict aerodynamic responses to arbitrary motions including pitch sinusoidal, chirp, Schroeder, and step. Model predictions were compared with actual CFD data. Overall, a good agreement was found for all models. Models to predict surface pressure data were also able to accurately predict the upper surface pressure data at different spanwise and chordwise locations at different angles of attack for both static and dynamic runs.

In §II. Theory, two reduced-order models are proposed, which the authors have termed: the quasi-steady model (§II.A. Quasi-steady one-dimensional model) & the inertial/hybrid model (§II.B. Quasi-one-dimensional pointmass model), respectively. N.b., in both cases time dependence isn’t explicitly modeled, i.e., technically speaking both models are quasi-steady. Ergo, in hindsight, it would have been more apt to call the model proposed in §II.A.: the matching-condition model. With that in mind, the readership is encouraged to substitute “matching-condition model/modeling regime” instead of “quasi-steady model/modeling regime,” when reading this conference paper. Moreover, the following title would have been more suitable: “Entropy-patch chokednozzle interaction: matching-condition and inertial modeling-regimes mapped”.

The underlying physical mechanism of the residual-based large eddy simulation (LES) based on the variational multiscale (VMS) method is clarified. Resolved large-scale energy transportation equation is mathematically derived for turbulent kinetic energy budget analysis. Firstly, statistical results of benchmark turbulent channel flow at Re_τ=180 obtained using a coarse mesh are compared with the results obtained by the classical LES with the Smagorinsky and dynamic subgrid stress (SGS) model. The present LES shows an advantage in predicting the statistical results of the incompressible turbulent flows. Secondly, the contributions of the unresolved small-scale presentation terms (Term I-IV in Eq. (10)) to the turbulent kinetic dissipation are analysed for the VMS method. The results show that the turbulent kinetic dissipation provided by the numerical diffusion in the VMS method is smaller in the inner layer, larger in the outer layer of the channel flow than those by the Smagorinsky and dynamic SGS model. The turbulent kinetic dissipation in the VMS method is mainly given by the numerical diffusion provided by one of the “cross-stress” terms (Term I, same as the stabilization term in the SUPG method) and LSIC term (Term IV). The other one of the “cross-stress” terms (Term II) gives rise to the positive turbulent kinetic energy budget, and does not dissipate the turbulent kinetic energy. The so-called “Reynolds stress” term (Term III) dissipates the turbulent energy but provides a very small numerical diffusion. Finally, on the basis of the turbulent kinetic energy dissipation analysis, a new residual-based stabilized finite element formulation is proposed by modifying the large-scale equation in the VMS method. Numerical experiments of 2D lid-driven cavity flow and 3D incompressible turbulent channel flow are tested to validate the proposed formulation. It is shown that all the stabilization terms in the proposed formulation produce additional numerical diffusions and physically increase the total turbulent kinetic dissipation. Consequently, an apparent improvement in both the first-order and second-order statistical quantities are pursued by the new stabilized finite element formulation.

Data-driven parameterizations offer considerable potential for improving the fidelity of General Circulation Models. However, ensuring that these remain consistent with the governing equations while still producing stable simulations remains a challenge. In this paper, we propose a combined Variational-Multiscale (VMS) Artificial Neural Network (ANN) discretization which makes no a priori assumptions on the model form, and is only restricted in its accuracy by the precision of the ANN. Using a simplified problem, we demonstrate that good predictions of the required closure terms can be obtained with relatively compact ANN architectures. We then turn our attention to the stability of the VMS-ANN discretization in the context of a single implicit time step. It is demonstrated that the ANN parameterization introduces nonphysical solutions to the governing equations that can significantly affect or prevent convergence. We show that enriching the training data with nonphysical states from intra-time step iterations is an effective remedy. This indicates that the lack of representative ANN-induced errors in our original, exact training inputs underpin the observed instabilities. In turn, this suggests that data set enrichment might aid in resolving instabilities that develop over several time steps.

Proper Orthogonal Decomposition (POD) plays an important role in the analysis of complex nonlinear systems governed by partial differential equations (PDEs), since it can describe the full-order system in a simplified but representative way using a handful of dominant dynamic modes. However, determining a POD from the results of complex unsteady simulations is often impractical using traditional approaches due to the need to store a large number of high-dimensional solutions. As an alternative, the incremental Singular Value Decomposition (SVD) has been developed, which can be used to avoid the storage problem by performing the POD analysis on the fly using a single-pass updating algorithm. Nevertheless, the total computing cost of incremental SVD is more than traditional approaches. In order to reduce this total cost, we incorporate POD mode truncation into the incremental procedure, leading to an enhanced algorithm for incremental SVD. Two error estimators are formulated for this enhanced incremental SVD based on an aggregated expression of the snapshot solutions, equipping the proposed algorithm with criteria for choosing the truncation number. The effectiveness of these estimators and the parallel efficiency of the enhanced algorithm are demonstrated using transient solutions from representative model problems. Numerical results show that the enhanced algorithm can significantly improve the computing efficiency for different kinds of datasets, and that the proposed algorithm is scalable in both the strong and weak sense.

An experimental cold-gas study of the response of a choked convergent–divergent nozzle to swirl perturbations is presented. The perturbations were obtained by means of upstream unsteady tangential injections into initially steady flows with different values of steady background swirl. The swirl perturbations induced changes in the axial mass-flow rate, due to either their ingestion or evacuation by the nozzle. This in turn caused a downstream acoustic response. For low-intensity background swirl the responses were found to be similar to those obtained without steady background swirl. Perturbations of a high-intensity background swirl led to different effects. For long injection times, the negative mass-flow rate modulation occurred in two stages. The first stage was similar to that of the background-swirl free case. The second stage occurred after a short time delay, and induced a much stronger negative acoustic response. This unexpected behavior suggests that a significant part of the tangentially injected fluid flows upstream inducing an accumulation of swirl, which is – after tangential injection is ceased – suddenly cleared out through the nozzle. A scaling rule for the amplitudes of these acoustic responses is reported. Furthermore, quasi-steady models, based on steady-state measurements are proposed. These models predict the downstream acoustic response amplitude within a factor two. Additionally, preliminary empirical evidence of the effect of swirl on the downstream acoustic response due to the interaction of entropy patches with a choked nozzle is reported. This was obtained by comparison of sound produced by abrupt radial or tangential sonic injection, upstream from the choked nozzle, of air from a reservoir at room temperature to that from a reservoir with a higher stagnation temperature. Because the mass flow through the nozzle does not increase instantaneously, the injected higher-enthalpy air accumulates upstream of the injection-port position in the main flow. This eventually induces a large downstream acoustic pulse when tangential injection is interrupted. The magnitude of the resulting sound pulse can reach that of a quasi-steady response of the nozzle to a large air patch with a uniform stagnation temperature equal to that of the upstream-injected heated air. This hypothesis is consistent with the fact that the initial indirect-sound pulse is identical to one obtained with unheated air injection. The authors posit that – given all of the insight gleaned from them in this case – acoustic measurements of indirect sound appear to be a potentially useful diagnostic tool.

Entropy inhomogeneities and vorticity spots induce so-called indirect combustion noise when passing through a choked nozzle; referred to as entropy noise and vorticity noise, respectively. We note that vorticity noise depends on the orientation of the vorticity; viz., oriented normal or parallel to the axial main flow. An experimental investigation of parallel component vorticity noise is presented. In the experiment a time-dependent swirling flow was induced by unsteady tangential injection in the pipe upstream of a choked convergent-divergent nozzle. As the resulting swirling flow passes through the nozzle, the axial stretching of the fluid caused an increase in rotation energy. The steady energy conservation in an isentropic flow implies a Mach number higher than unity at the throat and an associated reduction of density. Ergo, the critical mass-flow rate (for fixed reservoir pressure and temperature) decreases quadratically with increasing swirl intensity. The acoustic waves radiated downstream of the nozzle are due to the change in the mass flow through the nozzle. These are a direct measure for this mass-flow modulation. Using a semi-empirical model, this sound production mechanism is demonstrated to be quasi steady.

Adaptive Mesh Refinement (AMR) is potentially an effective way to automatically generate computational meshes for high-fidelity simulations such as Large Eddy Simulation (LES). When combined with adjoint methods, which are able to localize error contributions, AMR can generate meshes that are optimal for computing a physical quantity of interest (e.g. lift or drag). In order to apply adjoint-based AMR techniques to LES, primal flow solutions are needed to solve the adjoint problem backward in time. However, the resources required to store primal flow solutions can be huge, even prohibitive, in practical problems because of the typically very fine meshes and long averaging times for computing the statistical quantities of interest. Here, a Reduced-Order Representation (ROR) based upon proper orthogonal decomposition is introduced to address this issue. We develop an Enhanced Online Algorithm (EOA) based on incremental singular value decomposition to build this ROR online, which makes adjoint-based AMR feasible for practical applications. An adjoint-based error estimation procedure is first introduced, and verified using a manufactured solution. Then a ROR-driven AMR strategy is studied using a 1D unsteady Burgers problem with a multi-frequency forcing term. This is also used to evaluate the EOA for ROR-driven AMR. Numerical results demonstrate that the enhanced online algorithm generates RORs that are sufficiently accurate for AMR, avoiding the storage of almost all of the primal solution data.

Adaptive mesh refinement (AMR) is potentially an effective way to automatically generate computational meshes for high-fidelity simulations such as Large Eddy Simulation (LES). Adjoint methods, which are able to localize error contributions, can be used to optimize the mesh for computing a physical quantity of interest (e.g. lift, drag) during AMR. When adjoint-based AMR techniques are applied to LES, primal flow solutions are needed to solve the adjoint problem backward in time due to the nonlinearity of Navier-Stokes equations. However, the resources required to store primal flow solutions can be huge, even prohibitive, in practical problems because of the long averaging time for computing statistical quantities. In this paper, a Reduced-Order Model (ROM) based upon Proper Orthogonal Decomposition (POD) is introduced to circumvent this issue. First, an adjoint-based error estimation procedure is verified using a manufactured solution. Then a ROM-driven AMR strategy is studied using a LES model problem based on the 1D unsteady Burgers equation. Numerical results demonstrate that using ROMs not only lowers storage requirements, but also has no impact on the effectiveness of adjoint-based AMR.