Navier-Stokes Flow Research Articles

Comparison of Real and Simulation Aerodynamic Coefficients for 155 mm Ammunition Using Open-Source Code SU2 Software

In this study, density, energy, pressure, temperature changes, and drag coefficients occurring during the trajectory of movement for a 155 mm ammunition shell were simulated with the computational flow dynamics software Stanford University Unstructured (SU2). Reynolds-Averaged Navier-Stokes (RANS) equations, which are operationally simplified variations of the Navier-Stokes (N-S) flow solver, were used in the simulations. The Reynolds number based on the velocity was between 1.65x107 to 6.5x107 according to the Mach (M) number between 0.7 to 2.8 in the present simulation. The drag coefficients from 0.7 M to 2.8 M were obtained separately for each Mach 0.3 increase. The Free Computer-Aided Design (FreeCAD) program was used for geometrical drawings of the ammunition, the Geometry Description, Meshing, Solving, and Post-Processing (GMSH) software for mesh operations, and the Shear Stress Transport (SST) turbulence model to create a compressible finite volume. As well, 0 degrees was used as the angle of attack. For estimation of the aerodynamic coefficient percentage changes obtained from the simulations, exponential equations with the highest validity based on the R2 value were created. In addition, the calculated drag coefficients were compared with the actual values and a good fit was observed between them.

Sparsity-Preserving Two-Sided Iterative Algorithm for Riccati-Based Boundary Feedback Stabilization of the Incompressible Navier–Stokes Flow

In this paper, we explore the Riccati-based boundary feedback stabilization of the incompressible Navier–Stokes flow via the Krylov subspace techniques. Since the volume of data derived from the original models is gigantic, the feedback stabilization process through the Riccati equation is always infeasible. We apply aℋ2optimal model-order reduction scheme for reduced-order modeling, preserving the sparsity of the system. An extended form of the Krylov subspace-based two-sided iterative algorithm (TSIA) is implemented, where the computation of an equivalent Sylvester equation is included for minimizing the computation time and enhancing the stability of the reduced-order models with satisfying the Wilson conditions. Inverse projection approaches are applied to get the optimal feedback matrix from the reduced-order models. To validate the efficiency of the proposed techniques, transient behaviors of the target systems are observed incorporating the tabular and figurative comparisons with MATLAB simulations. Finally, to reveal the advancement of the proposed techniques, we compare our work with some existing works.

Existence and Uniqueness for Plane Stationary Navier–Stokes Flows with Compactly Supported Force

We study the stationary Navier--Stokes equations in the whole plane with a compactly supported force term and with a prescribed constant spatial limit. Prior to this work, existence of solutions to this problem was only known under special symmetry and smallness assumptions. In the paper we solve the key difficulties in applying Leray's {\it invading domains method} and, as a consequence, prove the existence of $D$-solutions in the whole plane for arbitrary compactly supported force. The boundary condition at infinity are verified in two different scenarios: (I) the~limiting velocity is sufficient large with respect to the external force, (II) both the total integral of force and the~limiting velocity vanish. Hence, our method produces large class of new solutions with prescribed spatial limits. Moreover, we show the uniqueness of $D$-solutions to this problem in a perturbative regime. The main tools here are two new estimates for general Navier-Stokes solutions, which have rather simple forms. They control the difference between mean values of the velocity over two concentric circles in terms of the Dirichlet integral in the annulus between them.

ელექტროგამტარ სითხეში ფოროვანი წრიული ფირფიტის ბრუნვით წარმოქმნილი სასაზღვრო ფენის სტაციონარული ამოცანა სუსტი მაგნიტური ველისა და თბოგადაცემის გათვალისწინებით ცვლადი გაჟონვის სიჩქარის შემთხვევაში

The paper investigates the stationary task of the boundary layer generated by the rotation of a circular porous plate rotating in an electrically conductive fluid with respect a weak magnetic field and heat transfer when the suction velocity into the plate changes by a linear law. The problem is solved by using the method of successive approximation (Shvec method). For solving it, in the magnetic field electrically conductive fluid flow Navier-Stokes and energy nonlinear differential equations in partial derivatives is used, which is recorded as a system of ordinary nonlinear differential equations using Karman general insertions. The solutions of the obtained system of equations are sought in the form of infinite series. The first two approaches are clearly found. Appropriate equations are obtained to determine the thickness of dynamic and thermal boundary layers and their exact solutions are obtained. All the kinematic characteristics of the fluid flow are calculated and images are obtained for the heat transfer and pressure distribution on the circular plate. The torque of rotation of the plate and the coefficient of heat transfer are also calculated. In the particular case, the relationship between the thicknesses of the dynamic and thermal boundary layers is established.

The Effect of Roll Rate on Simulated Entry Vehicle Ballistic Range Tests

Free-flight computational fluid/rigid-body dynamics simulations are now being explored as a way to augment physical experiments for characterizing the dynamic behavior of blunt-body atmospheric entry vehicles. Initializing these simulations with accurate initial conditions is critical to achieving validation against experimental results and characterizing vehicle behavior. This paper explores the impact of a nonnegligible initial roll rate on numerical free-flight simulations of ballistic range tests using the FUN3D Navier–Stokes flow solver and POST2 trajectory propagator. These ballistic range shots were performed on a model of the Supersonic Inflatable Aerodynamic Decelerator at the NASA Ames Hypervelocity Free-Flight Aerodynamics Facility. In the absence of measured roll data, a method for reconstructing the initial roll rate is developed assuming that a nonnegligible roll rate accounts for the exchange in amplitude between the experimental pitch and yaw data. Simulations of three shots spanning a range of initial roll rates are executed to evaluate the new method. Results are validated against the physical ballistic range tests. Simulations with the reconstructed roll rate are more accurate to the experimental data than those assuming a negligible initial roll rate. Direct calculation of the pitch damping coefficient also captures the effect of roll rate on pitching moment.

Attractors for the inhomogeneous incompressible Navier–Stokes flows

This paper is devoted to the evolution of Lions’s weak solutions to the inhomogeneous Navier–Stokes equations. After proving that the kinetic energy is eventually bounded, we obtain a weakly compact global attractor that all Lions’s weak solutions approach as time tends to infinity. Furthermore, the existence of attracting sets in strong topology is established for short trajectories satisfying an additional compactness condition on the density.

A generalized probabilistic learning approach for multi-fidelity uncertainty quantification in complex physical simulations

Two of the most significant challenges in uncertainty quantification pertain to the high computational cost for simulating complex physical models and the high dimension of the random inputs. In applications of practical interest, both of these problems are encountered, and standard methods either fail or are not feasible. To overcome the current limitations, we present a generalized formulation of a Bayesian multi-fidelity Monte-Carlo (BMFMC) framework that can exploit lower-fidelity model versions in a small data regime. The goal of our analysis is an efficient and accurate estimation of the complete probabilistic response for high-fidelity models. BMFMC circumvents the curse of dimensionality by learning the relationship between the outputs of a reference high-fidelity model and potentially several lower-fidelity models. While the continuous formulation is mathematically exact and independent of the low-fidelity model’s accuracy, we address challenges associated with the small data regime (i.e., only a small number of 50 to 300 high-fidelity model runs can be performed). Specifically, we complement the formulation with a set of informative input features at no extra cost. Despite the inaccurate and noisy information that some low-fidelity models provide, we demonstrate that accurate and certifiable estimates for the quantities of interest can be obtained for uncertainty quantification problems in high stochastic dimensions, with significantly fewer high-fidelity model runs than state-of-the-art methods for uncertainty quantification. We illustrate our approach by applying it to challenging numerical examples such as Navier–Stokes flow simulations and fluid–structure interaction problems.

Spatial Pointwise Behavior of Time-Periodic Navier–Stokes Flow Induced by Oscillation of a Moving Obstacle

We study the spatial decay of time-periodic Navier–Stokes flow at the rate |x|^{-1} with/without wake structure in 3D exterior domains when a rigid body moves periodically in time. In this regime the existence of time-periodic solutions was established first in the 2006 paper by Galdi and Silvestre, however, with little information about spatial behavior at infinity so that uniqueness of solutions was not available. This latter issue has been addressed by Galdi, who has recently succeeded in construction of a unique time-periodic solution with spatial behavior mentioned above if translational and angular velocities of the body fulfill, besides smallness and regularity, either of the following assumptions: (i) translation or rotation is absent; (ii) both velocities are parallel to the same constant vector. This paper shows the existence of a unique time-periodic Navier–Stokes flow in the small with values in the weak-L^3 space and then deduces the desired pointwise decay of the solution under some condition on the rigid motion of the body, that covers the cases (i), (ii) mentioned above.

Searching for Singularities in Navier–Stokes Flows Based on the Ladyzhenskaya–Prodi–Serrin Conditions

In this investigation we perform a systematic computational search for potential singularities in 3D Navier-Stokes flows based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that if the quantity $\int_0^T \| \mathbf{u}(t) \|_{L^q(\Omega)}^p \, dt$, where $2/p+3/q \le 1$, $q > 3$, is bounded, then the solution $\mathbf{u}(t)$ of the Navier-Stokes system is smooth on the interval $[0,T]$. In other words, if a singularity should occur at some time $t \in [0,T]$, then this quantity must be unbounded. We have probed this condition by studying a family of variational PDE optimization problems where initial conditions $\mathbf{u}_0$ are sought to maximize $\int_0^T \| \mathbf{u}(t) \|_{L^4(\Omega)}^8 \, dt$ for different $T$ subject to suitable constraints. These problems are solved numerically using a large-scale adjoint-based gradient approach. Even in the flows corresponding to the optimal initial conditions determined in this way no evidence has been found for singularity formation, which would be manifested by unbounded growth of $\| \mathbf{u}(t) \|_{L^4(\Omega)}$. However, the maximum enstrophy attained in these extreme flows scales in proportion to $\mathcal{E}_0^{3/2}$, the same as found by Kang et al. (2020) when maximizing the finite-time growth of enstrophy. In addition, we also consider sharpness of an a priori estimate on the time evolution of $\| \mathbf{u}(t) \|_{L^4(\Omega)}$ by solving another PDE optimization problem and demonstrate that the upper bound in this estimate could be improved.

Shape optimization of Navier–Stokes flows by a two-grid method

We consider the energy dissipation minimization constrained by steady Navier–Stokes flows. The nonlinearity of the Navier–Stokes equation causes its numerical solver computationally expensive and thus leads to inefficiency of shape optimization algorithms. We propose a new efficient shape gradient algorithm of distributed type by using a two-grid solver for the Navier–Stokes equation. Asymptotically optimal convergence rate is shown for mixed finite-element approximation of the two-grid distributed shape gradient. Moreover, a Uzawa iterative scheme for Stokes-type systems is adopted in the two-grid solver and a new scalar-type shape gradient flow is proposed to improve significantly the computational efficiency especially for 3D. Numerical experiments are presented to verify theoretical analysis and show effectiveness of two-grid shape gradient optimization algorithms.

Decay rates for the higher-order derivatives of the Navier-Stokes flow in exterior domains

This paper makes investigations on the time-decay rates for the 3-dimensional Navier-Stokes flow in exterior domains. Boundedness of the convolution operator with the Gaussian kernel from the intersection L01(Ω)∩L1(Ω,|x|αdx) for some 0<α≤1 to the space Lr(Ω) for all 1≤r<∞ is firstly established, from which, an innovated estimate ‖u(t)‖r≤Ct−α2−32(1−1r) (t≥T0, 1<r≤∞) is derived. Decay rates of the higher-order spatial and temporal derivatives of the flow are mainly studied then. In order to derive the estimates ‖∇∂tku(t)‖r≤Ct−k+12−α2−32(1−1r) adequately, translation transformation v˜kv(t)=∂tku(t+Tk) has been made.

Prediction of Rotating Instabilities in Low Pressure Steam Turbines Operating at Low Load

Abstract Due to the increasing need for operational flexibility nowadays, the low-pressure (LP) steam turbine might face widely variable operation conditions up to extreme low volume flow situations. Hereby, rotating instabilities (RIS), well known from compressor aerodynamics, can initiate nonsynchronous blade excitations, which can lead to increased vibrations of the last stage moving blade (L-0R). This paper describes a numerical method able to be used in an applicable way for blade designing. To judge the onset and the shape of RIS, a numerical approach based on a transient three-dimensional unsteady Reynolds averaged Navier–Stokes flow computation of the single passage L-1 stage, a full row L-0 stage, and an axis-symmetric diffuser was established. Mesh size, solver settings, etc., were optimized to reduce the computational effort without losing prediction accuracy. The dominant harmonic compound of a spatial Fourier transform of all monitored L-0R blade forces over the blade row's circumference reveals the stall cell count and excitation amplitude for each time-step. The transient change of the corresponding phase lag provided the stall cell speed and, multiplied with the stall cell count, the excitation frequency. Their distribution over decreasing flow rates finally displays the likely onset of RIS. The approach is capable to run the analyses of several load conditions within a couple of days and was successfully validated using measured blade tip deflections and unsteady pressure probes in multistage test turbines. Furthermore, it is shown that the approach can be used to find LP blade designs, where the effects of RIS can be significantly suppressed.

Numerical Simulation of Acoustic Resonance in a Duct Containing a Flat Plate

This work is about numerical simulations of vortex induced acoustic resonance in a duct, experimentally investigated by Welsh et al. Vortex shedding of low-speed flow over a flat plate excites an acoustic duct mode, leading to a lock-in of shedding and acoustic resonance frequency over a wide range of flow velocities. This study shows that a state-of-the-art compressible Navier–Stokes flow solver is able to capture the lock-in phenomenon. The focus is on the numerical parameters required to precisely recover the experimental results in terms of lock-in range and acoustic pressure levels. Complete and reliable physical data are thereby obtained, which can aid in developing a systematic understanding of the complex flow interactions. Furthermore, hysteresis behavior is discovered and numerically explored.

Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

For the Stokes system in the half space, Kang [Math. Ann. 331 (2005), pp. 87-109] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in an earlier paper. We also examine the Stokes flows generated by dipole bumps’ boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary.

Deformable model of a butterfly in motion on the example of Attacus atlas

Insect wings can undergo significant chordwise (camber) as well as spanwise (twist) deformation during flapping flight but the effect of these deformations is not well understood. The shape and size of butterfly wings leads to particularly large wing deformations, making them an ideal test case for investigation of these effects. High-speed videogrammetry was used to capture the wing kinematics and deformations. The movements of selected markers on the wings of a living insect was observed. Created characteristics showing the displacement in a three-dimensional coordinate system identified the kinematics and deformations of the butterfly's wings. These experimental results were then analyzed computationally using a high-fidelity, three-dimensional, unsteady Navier-Stokes flow solver. Computational fluid dynamics (CFD) simulations were carried out on the basis of the wing geometry of the large moth Attacus atlas. Six geometric and structural models of the Attacus atlas butterfly wing with various degrees of simplification were developed. Using these models, Fluid-Structure-Interaction (FSI) simulation studies were performed in the commercial Ansys software environment (Fluent and Mechanical). Computations of the wing beat cycle were carried out, obtaining pressure distributions, streamlines, vortex regions and cumulative force waveforms.

Asymptotic stability of stationary Navier–Stokes flow in Besov spaces

We discuss the asymptotic stability of stationary solutions to the incompressible Navier–Stokes equations on the whole space in Besov spaces. A critical estimate for the semigroup generated by the Laplacian with a perturbation is the main ingredient of the argument.

Efficient hyperreduction of high-order discontinuous Galerkin methods: Element-wise and point-wise reduced quadrature formulations

We develop and assess projection-based model reduction methods for high-order discontinuous Galerkin (DG) discretizations of parametrized nonlinear partial differential equations with applications in aerodynamics. Our emphasis is on the choice of hyperreduction methods. We analyze computational complexity and use numerical examples to show that typical hyperreduction methods based on element-wise or index-wise sampling of the residual vector cannot effectively reduce high-order DG discretizations, with a large number of degrees of freedom and quadrature points per element and conversely a small number of elements. To overcome this limitation, we devise a hyperreduction method for high-order DG methods that samples individual quadrature points, instead of elements, to provide a finer decomposition of the residual. We compare the formulation, implementation, and computational complexity of the element-wise and point-wise formulations. We then numerically assess the two formulations using Euler flow over a shape-parametrized airfoil and laminar and turbulent Navier-Stokes flow over a three-dimensional aerodynamic body. In each case, we observe that the point-wise hyperreduced model reduces the computational time by several orders of magnitude relative to the high-order DG method and a few orders of magnitude relative to the element-wise hyperreduced model. In addition, the computational performance of the point-wise formulation does not deteriorate with the DG polynomial degree, unlike for the element-wise formulation.

Dynamics for 2D incompressible Navier–Stokes flow coupled with time-dependent Darcy flow

In this paper, we use the method of evolutionary systems introduced by Cheskidov and Foias to describe the existence of global attractor for 2D incompressible Navier–Stokes flow coupled with time-dependent Darcy flow. Furthermore, stationary statistical solutions of this system are constructed from the global attractor.

Some periodic orbits of chaotic motions for time-periodic forced two-dimensional Navier–Stokes flows

In this study, we study the two-dimensional Navier–Stokes flows with time-periodic external forces. Invariant solutions, including periodic orbits and relative periodic orbits, are extracted with the recurrent flow analysis, while low-dimensional projections based on the dynamic mode decomposition algorithm are used to reduce the cost of searching nearly recurrences. When the period of forces gets a constant increase, the flows change from the stable time-periodic state to oscillate and even turbulent flows. In all cases, one periodic orbit is identified near the initial stage. This orbit represents the stable/unstable base state, and the trajectories of vorticity fields are trapped inside it or escape away from it leading to oscillating/turbulent motions. For the oscillating flows, periodic orbits without any symmetries play the role that the flows visit them and then move away from them to other orbits. In addition, for a moderate period of forces, a bursting phenomenon occurs and the state of oscillating flows turns to turbulent flows with the rapid increase in energy. For the turbulent motions, one unstable periodic, which qualitatively represents the shapes of a large vortex dipole that exists in the turbulent motions, is obtained. Its statistical significance is shown by the frequency of that flows visit it.

A shape design problem for the Navier–Stokes flow with a convective boundary condition

In this study, a shape optimization problem for the two-dimensional stationary Navier–Stokes equations with an artificial boundary condition is considered. The fluid is assumed to be flowing through a rectangular channel, and the artificial boundary condition is formulated so as to take into account the possibility of ill-posedness caused by the usual do-nothing boundary condition. The goal of the optimization problem is to maximize the vorticity of the said fluid by determining the shape of an obstacle inside the channel. Meanwhile, the shape variation is limited by a perimeter functional and a volume constraint. The perimeter functional was considered to act as a Tikhonov regularizer and the volume constraint is added to exempt us from possible topological changes. The shape derivative of the objective functional was formulated using the rearrangement method, and this derivative was later on used for gradient descent methods. Additionally, an augmented Lagrangian method and a class of solenoidal deformation fields were considered to take into account the goal of volume preservation. Lastly, numerical examples based on gradient descent and the volume preservation methods are presented.