Linearized Navier-Stokes Equations Research Articles

Role of Klebanoff modes in active flow control of separation: direct numerical simulations

Our previous research has shown that an active flow control strategy using two-dimensional (2-D) harmonic blowing and suction with properly chosen frequency and amplitude can significantly reduce the separation region, delay transition to turbulence and can even relaminarize the flow. How such effective flow control for transition delay and relaminarization is affected by free-stream turbulence (FST) remains an open question. In order to answer this question, highly resolved direct numerical simulations (DNS) are carried out where very low-amplitude isotropic FST fluctuations are introduced at the inflow boundary of the computational domain. With FST the effectiveness of the flow control is not diminished, and the extent of the separated flow region is reduced by the same amount as for the zero FST case. However, a striking difference observed in the DNS is the fact that in the presence of even very low levels of FST, the flow transitions shortly downstream of the reattachment location of the bubble, contrary to the case without FST. It appears that this different behaviour for even very small levels of FST is caused by an interaction between the high-amplitude 2-D disturbances introduced by the flow control forcing and 3-D Klebanoff modes (K-modes) that are generated by the FST. The streamwise elongated streaks due to the K-modes cause a spanwise-periodic modulation of the basic flow and subsequently of the primary 2-D wave. The disturbances associated with this modulation exhibit strong growth and initiate the breakdown process to turbulence. Linear secondary instability investigations with respect to low-frequency 3-D disturbances are carried out based on the linearized Navier–Stokes equations. The response of the forced flow to the low-frequency 3-D disturbances reveals that the time-periodic base flow is unstable with respect to a wide range of 3-D modes. In particular, the wavelength associated with the spanwise spacing of the K-mode falls into the range of, and is in fact very close to, the most unstable 3-D disturbances. Results from the secondary instability analysis and the comparison with DNS results, support the conjecture that the forcing amplitude has a major impact on the onset and amplification rate of the K-modes: lowering the forcing amplitude postpones the onset of the growth of the K-modes and reduces the growth rate of the K-modes for a given FST intensity. The net effect of these two events is a delay of the transition onset. Nevertheless, the instability mechanism that governs the transition process is the same as previously identified, i.e. interaction of the K-mode and 2-D primary wave. Furthermore, for low levels of FST, the amplitude of the low-frequency K-modes scales linearly with the FST intensity in the approach boundary layer up to the secondary instability regime.

Thermoacoustic analysis of a laminar premixed flame using a linearized reactive flow solver

In this paper, the dynamics and thermoacoustic stability of a laminar premixed flame are analyzed using a linearized reactive flow (LRF) solver. The LRF solver is based on linearized compressible Navier-Stokes and reacting species transport equations and thereby includes a model for the dynamic response of the flame to flow perturbations in an inherent manner. The equations are discretized using the discontinuous Galerkin finite element method. By way of example, thermoacoustic characteristics of attached and lifted laminar premixed flames are investigated. First, the respective flame transfer functions (FTFs) are computed in the frequency domain with the LRF solver. The results are in agreement with reference FTFs identified from CFD time-series. Secondly, the LRF solver is employed for thermoacoustic stability analysis, i.e. computation of shape, frequency, and growth rate of eigenmodes. Results are compared to established hybrid methods that couple FTFs with a low-order thermoacoustic network-model or a linearized Navier-Stokes equations solver. All solvers capture the dominant thermoacoustic mode, but only the LRF resolves local flow-flame interaction, revealing e.g. the onset of the flame movement and the propagation of distortions along the flame.

On Numerical Simulation of Sound Damping Mechanisms in the Cell of a Sound-Absorbing Structure

Numerical simulation of acoustic processes with an interferometer at high acoustic pressure levels is one of the ways to noise reduction processes using samples of sound-absorbing structures (SAS). In the course of research, an SAS sample consisting of a single Helmholtz resonator of circular shape was used. Studies were conducted at sound pressure levels of 110, 130, 140, and 150 dB. Results obtained with the interferometer with normal wave incidence were taken as a basis. In the calculations, systems of linearized and complete Navier–Stokes equations were used. The results obtained with the linearized Navier–Stokes equations make it possible to determine the acoustic characteristics of the sample for linear modes of operation with sufficient accuracy. The complete system of Navier–Stokes equations, taking into account compressibility, allowed good qualitative and quantitative agreement with the experiment at high acoustic pressure levels.

Observer-based feedback boundary stabilization of the Navier–Stokes equations

This paper aims at designing an observer-based feedback law which locally stabilizes the solution to the two dimensional Navier–Stokes equations with mixed boundary conditions. We consider a finite number of controls acting on a portion of the boundary through Robin boundary conditions and construct a linear Luenberger observer based on the point observations of the linearized Navier–Stokes equations. The sensor location for the point observations is determined by the response of feedback functional gains. We prove that the nonlinear system coupled with the observer through the feedback law is locally exponentially stable. Numerical experiments based on a Taylor–Hood finite element method are presented to illustrate the design for different Reynolds numbers.

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.

Heat and mass transfer in a rarefied gas confined between its two parallel condensed phases

The heat and mass transfer in a rarefied gas between its two parallel condensed phases is considered on the basis of linearized and non-linear S-model kinetic equations. The profiles of the macroscopic parameters in the gap between gas-liquid interfaces are obtained for several Knudsen numbers and for the cases of complete and non-complete evaporation and condensation. The linearized Navier–Stokes equations and energy equation, subjected to the temperature and pressure jump boundary conditions, are solved analytically and the expressions for the macroscopic parameters are obtained. The comparison of three approaches allowed us to establish the limit of the application of the linearized approaches in term of the saturation temperature ratio between the condensed phases.

Acoustic receptivity and transition modeling of Tollmien-Schlichting disturbances induced by distributed surface roughness

Acoustic receptivity to Tollmien-Schlichting waves in the presence of surface roughness is investigated for a flat plate boundary layer using the time-harmonic incompressible linearized Navier-Stokes equations. It is shown to be an accurate and efficient means of predicting receptivity amplitudes and, therefore, to be more suitable for parametric investigations than other approaches with direct-numerical-simulation-like accuracy. Comparison with the literature provides strong evidence of the correctness of the approach, including the ability to quantify non-parallel flow effects. These effects are found to be small for the efficiency function over a wide range of frequencies and local Reynolds numbers. In the presence of a two-dimensional wavy-wall, non-parallel flow effects are quite significant, producing both wavenumber detuning and an increase in maximum amplitude. However, a smaller influence is observed when considering an oblique Tollmien-Schlichting wave. This is explained by considering the non-parallel effects on receptivity and on linear growth which may, under certain conditions, cancel each other out. Ultimately, we undertake a Monte Carlo type uncertainty quantification analysis with two-dimensional distributed random roughness. Its power spectral density (PSD) is assumed to follow a power law with an associated uncertainty following a probabilistic Gaussian distribution. The effects of the acoustic frequency over the mean amplitude of the generated two-dimensional Tollmien-Schlichting waves are studied. A strong dependence on the mean PSD shape is observed and discussed according to the basic resonance mechanisms leading to receptivity. The growth of Tollmien-Schlichting waves is predicted with non-linear parabolized stability equations computations to assess the effects of stochasticity in transition location.

Analytical solutions for conformable space-time fractional partial differential equations via fractional differential transform

This paper introduces an efficient fractional differential transform that is called “conformable fractional partial differential transform (CFPDT)” and its properties for solving linear and nonlinear conformable space-time fractional partial differential equations (CSTFPDEs). Moreover, a CFPDT is more practical and helpful for solving abroad CSTFPDEs. Analytical solutions to linear Navier–Stokes equation and nonlinear gas dynamic equations in sense of conformable space-time fractional partial derivatives are successfully obtained to confirm the accuracy and efficiency of the proposed transform.

Linear nonreciprocal transmission of sound in a viscous fluid with asymmetric scattteres

Reciprocity theorem is hold for wave equation due to time-reversal symmetry. Dissipative losses introduced by adding imaginary part to elastic modulus make sound propagation irreversible but the reciprocity theorem remains valid. In viscous fluid acoustic oscillations of local velocity follow the Navier-Stokes equation, which is not time-reversible. However, broken T-symmetry does not necessarily lead to nonreciprocity. Here we demonstrate that the necessary condition for nonreciprocal propagation in a viscous fluid is broken P-symmetry, which can be achieved by introducing asymmetric scatterers in a viscous environment. The experiment was done with a phononic crystal of Al rods in water. The rods have asymmetric cross-section in a form of a circular sector with an arc of 120 degrees. The measured transmission spectrum exhibits signatures of nonreciprocity within a range of frequencies from 300 to 450 kHz. The experimental results are in agreement with numerical simulations based on the linearized Navier-Stokes equation. The nonrecipocity is due to different viscous losses accumulated along sound propagation in two opposite directions with broken P-symmetry. Nonreciprocity increases for the rods with more rough surfaces since dissipation occurs mainly in a viscous boundary layer of thickness of few microns. Unlike previously proposed nonreciprocal devices based on nonlinearity or local spinning of fluid, our passive device is small, cheap, and does not require energy source. [Work supported by the NSF under Grant No. 1741677.]

Computation of the bluff-body sound generation by a self-consistent mean flow formulation

The sound generated by the flow around a circular cylinder is numerically investigated by using a finite-element method. In particular, we study the acoustic emissions generated by the flow past the bluff body at low Mach and Reynolds numbers. We perform a global stability analysis by using the compressible linearized Navier-Stokes equations. The resulting direct global mode provides detailed information related to the underlying hydrodynamic instability and data on the acoustic field generated. In order to recover the intensity of the produced sound, we apply the self-consistent model for non-linear saturation proposed by Mantič-Lugo, Arratia, and Gallaire [“Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake,” Phys. Rev. Lett. 113, 084501 (2014)]. The application of this model allows us to compute the amplitude of the resulting linear mode and the effects of saturation on the mode structure and acoustic field. Our results show excellent agreement with those obtained by a full compressible simulation direct numerical simulation and those derived by the application of classical acoustic analogy formulations.

The Effect of Viscosity of a Fluid on the Frequency Response of a Viscoelastic Plate Loaded by This Fluid

A hydroviscoelastic system consisting of a viscoelastic plate and a half-plane filled with a viscous fluid is considered. The effect of viscosity of the fluid on the frequency response of the system and its dependence on the rheological parameters of plate material are estimated. The problem on forced vibrations of the system in the plane strain state is investigated using the exact equations of viscoelastodynamics for describing the motion of the plate and linearized Navier-Stokes equations for describing the flow of the fluid. The results found in the cases of nonviscous compressible and Newtonian compressible viscous fluids are compared.

Wave load on a navigation lock sliding gate

The wave load on a navigation lock sliding gate equipped with a ballast tank is investigated assuming the linear wave theory is valid and using the eigenfunction expansion matching method to solve the governing equations. The correctness and the limits of the solution have been checked by comparing the results with those of a numerical code which solve the full non linear Navier-Stokes equations. The results relating to the effect of the geometrical parameters and wave characteristics on the load acting on the gate are presented and discussed.The ballast tank is found to increase the complexity of the phenomenon in relation to the wave interaction with the gate. The results indicate that the peak value of the vertical force occurs for wave numbers mostly dependent on the tank length and on the tank position along the depth, while the thickness has a smaller influence. The ballast tank has also a significant effect on the horizontal dynamic force acting on the gate, which vanishes when the wave number takes particular values. Finally, the moment applied to the gate shows a dependency on the geometrical and hydrodynamic parameters similar to that of the forces.

Efficient aeroelastic reduced order model with global structural modifications

Time domain aeroelastic analysis has high computing costs when using computational fluid dynamics. These costs become prohibitive when the structural model undergoes large changes from the baseline design, as within an aircraft design process. To overcome this realistic challenge, we have developed, implemented, and demonstrated an efficient method that is robust in the presence of global modifications of the structure. The method consists of: a) a reduced order model of the linearized Navier–Stokes equations generated around an aeroelastic equilibrium that depends, in turn, on the structural model; b) an approximate structural dynamic reanalysis method valid for global modifications of the structure; and c) a mechanism to exchange information between fluid and structural solvers without need for calculating at each iteration of the structural design an eigenvalue problem of the modified structure. The resulting aeroelastic reduced order model is demonstrated for the AGARD 445.6 wing, and material properties are varied up to 100% from their original values. It is found that: a) predictions of the time domain aeroelastic response and of the flutter speed are accurate for all modifications of the structure; and b) the computational efficiency of the proposed aeroelastic reduced order model is linearly proportional to the number of structural configurations considered. The method, therefore, is ideally suited for optimization and uncertainty studies.

Linearized Navier–Stokes equations for aeroacoustics using stabilized finite elements: Boundary conditions and industrial application to aft-fan noise propagation

In this paper, a numerical method for solving the linearized Navier-Stokes equations is presented for aeroacoustic sound propagation problem. The Navier–Stokes equations are linearized in the frequency domain. The fan noise of jet engine is emitted nearly selectively at certain frequencies, which depend on the rotation velocity of the fan. A frequency domain approach is highly suitable for this kind of problem, instead of a costly time-dependent simulation which can handle a large range of frequencies depending on the time step and the mesh.The calculations presented here were all made using Aether, a Navier–Stokes code which uses finite elements stabilized with SUPG (Streamline Upwind Galerkin). Automatic code differentiation was used to linearize this code. Entropy variables bring interesting mathematical properties to the numerical scheme, but also prevent the easy implementation of boundary conditions. For instance, the pressure is a non-linear combination of the entropy variables. Imposing a pressure variation needs a linearization of this relation which is detailed herein. The performance of different types of boundary conditions used to impose the acoustic pressure variation inside the engine is studied in detail. Finally, a very surprising effect of the SUPG scheme was to transform a homogeneous Dirichlet boundary condition on all variables to a transparent one which is able to let only outgoing waves pass through with no incoming wave. A one-dimensional toy model is given to explain how SUPG brings about this transformation.The last part of the article is dedicated to an industrial test case. The geometry of a model turbine from the Clean Sky European project was used for sound propagation of the fan exhaust noise of a jet engine. Computations on several modes with increasing complexities were done and the results compared to a boundary element method which served as a reference when no mean flow is present. Results of a computation with a mean flow are shown.

Analysis and modeling of localized invariant solutions in pipe flow

Turbulent spots surrounded by laminar flow are a landmark of transitional shear flows, but the dependence of their kinematic properties on spatial structure is poorly understood. We here investigate this dependence in pipe flow for Reynolds numbers between 1500 and 5000. We compute spatially localized relative periodic orbits in long pipes and show that their upstream and downstream fronts decay exponentially towards the laminar profile. This allows to model the fronts by employing the linearized Navier-Stokes equations, and the resulting model yields the spatial decay rate and the front velocity profiles of the periodic orbits as a function of Reynolds number, azimuthal wave number and propagation speed. In addition, when applied to a localized turbulent puff, the model is shown to accurately approximate the spatial decay rate of its upstream and downstream tails. Our study provides insight into the relationship between the kinematics and spatial structure of localized turbulence and more generally into the physics of localization.

Diffusion Processes in the A-Model of Vector Admixture: Turbulent Prandtl Number

Using analytical approach of the field theoretic renormalization-group technique in two-loop approximation we model a fully developed turbulent system with vector characteristics driven by stochastic Navier-Stokes equation. The behaviour of the turbulent Prandtl number PrA,tis investigated as a function of parameterAand spatial dimensiond> 2 for three cases, namely, kinematic MHD turbulence (A= 1), the admixture of a vector impurity by the Navier-Stokes turbulent flow (A= 0) and the model of linearized Navier-Stokes equation (A= −1). It is shown that forA= −1 the turbulent Prandtl number is given already in the one-loop approximation and does not depend ondwhile turbulent Prandt numbers in first two cases show very similar behaviour as functions of dimensiondin the two-loop approximation.

The influence of the initial strains of the highly elastic plate on the forced vibration of the hydro-elastic system consisting of this plate, compressible viscous fluid, and rigid wall

The hydro-elastic system consisting of a pre-stretched highly elastic plate, compressible Newtonian viscous fluid, and the rigid wall is considered and it is assumed that on the plate a lineal-located time-harmonic force acts. It is required to investigate the dynamic behavior of this system and determine how the problem parameters and especially the pre-straining of the plate acts on this behavior. The elasticity relations of the plate are described through the harmonic potential and linearized (with respect to perturbations caused by external time-harmonic force) form of these relations is used in the present investigation. The plane-strain state in the plate is considered and the motion of that is described within the scope of the three-dimensional linearized equations of elastic waves in elastic bodies with initial stresses. The motion of the fluid is described by the linearized Navier-Stokes equations and it is considered the plane-parallel flow of this fluid. The Fourier transform with respect to the space coordinate is applied for a solution to the corresponding boundary-value problem. Numerical results on the frequency response of the interface normal stress and normal velocity and the influence of the initial stretching of the plate on this response are presented and discussed. In particular, it is established that the initial stretching of the plate can decrease significantly the absolute values of the aforementioned quantities.

Frequency-dependent hydrodynamic interaction between two solid spheres

Hydrodynamic interactions play an important role in many areas of soft matter science. In simulations with implicit solvent, various techniques such as Brownian or Stokesian dynamics explicitly include hydrodynamic interactions a posteriori by using hydrodynamic diffusion tensors derived from the Stokes equation. However, this equation assumes the interaction to be instantaneous which is an idealized approximation and only valid on long time scales. In the present paper, we go one step further and analyze the time-dependence of hydrodynamic interactions between finite-sized particles in a compressible fluid on the basis of the linearized Navier-Stokes equation. The theoretical results show that at high frequencies, the compressibility of the fluid has a significant impact on the frequency-dependent pair interactions. The predictions of hydrodynamic theory are compared to molecular dynamics simulations of two nanocolloids in a Lennard-Jones fluid. For this system, we reconstruct memory functions by extending the inverse Volterra technique. The simulation data agree very well with the theory, therefore, the theory can be used to implement dynamically consistent hydrodynamic interactions in the increasingly popular field of non-Markovian modeling.

A Numerical Model of an Acoustic Metamaterial Using the Boundary Element Method Including Viscous and Thermal Losses

In recent years, boundary element method (BEM) and finite element method (FEM) implementations of acoustics in fluids with viscous and thermal losses have been developed. They are based on the linearized Navier–Stokes equations with no flow. In this paper, such models with acoustic losses are applied to an acoustic metamaterial. Metamaterials are structures formed by smaller, usually periodic, units showing remarkable physical properties when observed as a whole. Acoustic losses are relevant in metamaterials in the millimeter scale. In addition, their geometry is intricate and challenging for numerical implementation. The results are compared with existing measurements.

Flow and Viscous Effects on Impedance Eduction

In principle, the impedance of a locally reacting acoustic liner should be independent of the structure of the sound field above the liner. Some experimental studies, however, have found that the impedance inversely determined by available eduction techniques is actually dependent on the direction of incident waves relative to the mean flow. This dependence has mainly been attributed to the incomplete description of flow physics in the eduction techniques, particularly to the neglect of the viscous effects. The purpose of the paper is to examine whether such a wave-direction dependence can be removed by including the flow physics that is absent in previous eduction techniques. To this end, two eduction methods using the full linearized Navier–Stokes equations are proposed. Possible turbulent effects are considered by using available turbulence models. The proposed eduction methods are applied to new measurement data and data available in the literature, and are compared with another eduction method that has been commonly used in the community. The results show that including viscous effects does not resolve the controversy on the apparent wave-direction dependence.