
[Sponsors] 
Publication date: Available online 20 September 2022
Source: Computers & Fluids
Author(s): Y. Mehta, R.J. Goetsch, O.V. Vasilyev, J.D. Regele
Publication date: Available online 20 September 2022
Source: Computers & Fluids
Author(s): Zhicheng Wang, Muzammil Soomro, Cheng Peng, Luis F. Ayala, Orlando M. Ayala
Publication date: Available online 22 September 2022
Source: Computers & Fluids
Author(s): Kyle A. Schau, Chelsea Johnson, Julia Muller, Joseph C. Oefelein
Publication date: Available online 22 September 2022
Source: Computers & Fluids
Author(s): Hemanth Chandra Vamsi Kakumani, Nagabhushana Rao Vadlamani, Paul Gary Tucker
Publication date: Available online 22 September 2022
Source: Computers & Fluids
Author(s): Jiangxu Huang, Lei Wang, Kun He
Publication date: Available online 17 September 2022
Source: Computers & Fluids
Author(s): Chao Ma, Jie Wu, Liming Yang
Publication date: Available online 17 September 2022
Source: Computers & Fluids
Author(s): Rodrigo L.F. Castello Branco, Bruno B.M. Kassar, João N.E. Carneiro, Angela O. Nieckele
Publication date: Available online 3 September 2022
Source: Computers & Fluids
Author(s): Kazuya Kusano
Publication date: Available online 13 September 2022
Source: Computers & Fluids
Author(s): A. Bußmann, J. Buchmeier, M.S. Dodd, S. Adami, I. BermejoMoreno
Publication date: Available online 16 September 2022
Source: Computers & Fluids
Author(s): Guoxiang Grayson Tong, David Kamensky, John A. Evans
Studies on the unphysical increase of turbulent quantities for RANS simulation induced by shock waves in hypersonic flows are carried out. Numerical experiments on the hypersonic flow over a blunt body reveal that the phenomenon of unphysical increase of turbulent quantities across the detached shock wave is induced by the strainratebased production terms of the kw and kw SST turbulence models, which leads to the overprediction of aerothermal prediction. While this phenomenon does not occur for SpalartAllmaras (SA) turbulence model because of its vorticitybased production term. In order to eliminate this unphysical phenomenon, and to maintain the accuracy of the original models for boundary layer and separation flows, a new correction method for the kw and kw SST models is proposed: by comparing the orders of magnitude between the strainratebased and vorticitybased production terms, the vorticitybased production term is used near the shock waves, while the original strainratebased production term is still used in other regions. Finally, the correction method is applied to turbulence and transition flows over blunt bodies, and the numerical results show that the correction method effectively eliminates the unphysical increase of turbulent quantities across shock waves and improves the accuracy of aerothermal and transition onset location prediction.
A numerical study of the fused deposition modeling (FDM) process using a boundaryconforming freesurface finite element approach is performed. Due to the complexity of the FDM process, among all of its parts, we focus on the deposition and spreading of an individual filament. The polymer behavior, i.e., the shear rate dependent and temperaturedependent viscosity, is included by the CrossWLF viscosity model. The moving domain is addressed by the virtual region mesh update method, which, in the present paper, is extended to freesurface problems. The particularity of dividing the mesh domain into an activated and a deactivated domain makes it possible to handle large translatory mesh deformation. In this work, we make use of the level of detail offered by a boundaryconforming approach regarding both topology accuracy and the imposition of boundary conditions in order to study the deposition of a single filament at a small scale. Parameters with a direct impact on the mechanical properties of the final object can be straightforwardly computed by a boundaryconforming approach, for instance, the crosssection, the contact area, the temperature distribution, and the heat fluxes over the surfaces. The presented approach is validated by a twodimensional benchmark test case before the numerical results of the threedimensional simulation of the filament deposition are shown.
A set of coupling equations to appropriately couple nonlinear 1D and lumpedparameter (0D) models for blood flow in compliant vessels is defined. Then, a methodology for the highorder numerical coupling between 1D and 0D vessels through hybrid junctions is proposed. Finally, an effective methodology to construct hybrid 1D0D networks of vessels is developed, where different apriori model selection criteria are explored, focusing on obtaining the best possible tradeoff between computational cost of the simulations and accuracy of the computed solutions for the hybrid network with respect to the reference 1D network.
The onedimensional (1D) modeling of blood flow in complex networks of vessels and cardiovascular models can result in computationally expensive simulations. The complexity of such networks has significantly increased in the last years, in terms of both enhanced anatomical detail and modeling of physiological mechanisms and mechanical characteristics. To address such issue, the main goal of this work is to present a novel methodology to construct hybrid networks of coupled 1D and 0D vessels and to perform computationally efficient and accurate blood flow simulations in such networks. Departing from both the 1D and lumpedparameter (0D) nonlinear models for blood flow, we propose highorder numerical coupling strategies to solve the 1D, 0D, and hybrid coupling of vessels at junctions. To effectively construct hybrid networks, we explore different apriori model selection criteria focusing in obtaining the best possible tradeoff between computational cost of the simulations and accuracy of the computed solutions for the hybrid network with respect to the 1D network. The achievement of the expected order of accuracy is verified in several test cases. The novel methodology is applied to two different arterial networks, the 37artery network and the reduced ADAN56 model, where, in order to identify the best performing apriori model selection criteria, the quantitative assessment of CPU times and errors and the qualitative comparison between results are carried out and discussed.
Riemann problem for compressible noslip driftflux model with modified Chaplygin twophase flows is investigated. The exact Riemann solution is also established through the characteristic analysis of compressible gasliquid flow. The compete wave structure is validated numerically through test cases selected form the open literature showing the main twophase flow variables.
In this paper, we concern about the Riemann problem for compressible noslip driftflux model which represents a system of quasilinear partial differential equations derived by averaging the mass and momentum conservation laws with modified Chaplygin twophase flows. We obtain the exact solution of Riemann problem by elaborately analyzing characteristic fields and discuss the elementary waves namely, shock wave, rarefaction wave and contact discontinuity wave. By employing the equality of pressure and velocity across the middle characteristic field, two nonlinear algebraic equations with two unknowns as gas density ahead and behind the middle wave are formed. The Newton–Raphson method of two variables is applied to find the unknowns with a series of initial data from the literature. Finally, the exact solution for the physical quantities such as gas density, liquid density, velocity, and pressure are illustrated graphically.
When simulating multiphase compressible flows using the diffuseinterface methods, important issues arise during the interaction of waves with diffused interfaces. A solution is to use a pressuredisequilibrium model with finite, instead of infinite, pressurerelaxation rate. A new numerical method to solve this model is proposed and solutions of the new modelling are examined and compared to literature for different problems and in particular for the study of a shock on a water–air interface.
When simulating multiphase compressible flows using the diffuseinterface methods, the test cases presented in the literature to validate the modellings with regard to interface problems are always textbook cases: interfaces are sharp and the simulations therefore easily converge to the exact solutions. In real problems, it is rather different because the waves encounter moving interfaces which consequently have already undergone the effects of numerical diffusion. Numerical solutions resulting from the interactions of waves with diffused interfaces have never been precisely investigated and for good reasons, the results obtained are extremely dependent on the model used. Precisely, wellposed models present similar and important issues when such an interaction occurs, coming from the appearance of a wavetrapping phenomenon. To circumvent those issues, we propose to use a thermodynamicallyconsistent pressuredisequilibrium model with finite, instead of infinite, pressurerelaxation rate to overcome the difficulties inherent in the computation of these interactions. Because the original method to solve this model only enables infinite relaxation, we propose a new numerical method allowing infinite as well as finite relaxation rates. Solutions of the new modeling are examined and compared to literature, in particular we propose the study of a shock on a water–air interface, but also for problems of helium–air and water–air shock tubes, spherical and nonspherical bubble collapses.
The transformthensolve approach is extended to standard finite element discretizations of the Stokes problem. The extension exploits two complementary strategies to limit the impact of the complexity increase due to the transformation. The comparison with a stateoftheart block diagonal preconditioner demonstrates that the transformthensolve approach is always competitive while being significantly more robust for problems with open flow boundary condition andor variable viscosity.
We consider numerical solution of finite element discretizations of the Stokes problem. We focus on the transformthensolve approach, which amounts to first apply a specific algebraic transformation to the linear system of equations arising from the discretization, and then solve the transformed system with an algebraic multigrid method. The approach has recently been applied to finite difference discretizations of the Stokes problem with constant viscosity, and has recommended itself as a robust and competitive solution method. In this work, we examine the extension of the approach to standard finite element discretizations of the Stokes problem, including problems with variable viscosity. The extension relies, on one hand, on the use of the successive overrelaxation method as a multigrid smoother for some finite element schemes. On the other hand, we present strategies that allow us to limit the complexity increase induced by the transformation. Numerical experiments show that for stationary problems our method is competitive compared to a reference solver based on a block diagonal preconditioner and MINRES, and suggest that the transformthensolve approach is also more robust. In particular, for problems with variable viscosity, the transformthensolve approach demonstrates significant speedup with respect to the block diagonal preconditioner. The method is also particularly robust for timedependent problems whatever the time step size.
In this article, a novel mass and momentum conservative semiimplicit finite volume scheme is developed for the coupled solution of hydrostatic shallow water flow and the movement of one or more floating rigid bodies. The coupling is achieved via a nonlinear volume function in the mass conservation equation that depends on the coordinates of the floating objects, whose dynamics is computed by solving a set of ordinary differential equations for their six degrees of freedom. The proposed algorithm may be useful for hydraulic engineering, such as for the simulation of ships moving in inland waterways and coastal regions.
Simulating fluidstructure interaction problems usually requires a considerable computational effort. In this article, a novel semiimplicit finite volume scheme is developed for the coupled solution of free surface shallow water flow and the movement of one or more floating rigid structures. The model is wellsuited for geophysical flows, as it is based on the hydrostatic pressure assumption and the shallow water equations. The coupling is achieved via a nonlinear volume function in the mass conservation equation that depends on the coordinates of the floating structures. Furthermore, the nonlinear volume function allows for the simultaneous existence of wet, dry and pressurized cells in the computational domain. The resulting mildly nonlinear pressure system is solved using a nested Newton method. The accuracy of the volume computation is improved by using a subgrid, and time accuracy is increased via the application of the theta method. Additionally, mass is always conserved to machine precision. At each time step, the volume function is updated in each cell according to the position of the floating objects, whose dynamics is computed by solving a set of ordinary differential equations for their six degrees of freedom. The simulated moving objects may for example represent ships, and the forces considered here are simply gravity and the hydrostatic pressure on the hull. For a set of test cases, the model has been applied and compared with available exact solutions to verify the correctness and accuracy of the proposed algorithm. The model is able to treat fluidstructure interaction in the context of hydrostatic geophysical free surface flows in an efficient and flexible way, and the employed nested Newton method rapidly converges to a solution. The proposed algorithm may be useful for hydraulic engineering, such as for the simulation of ships moving in inland waterways and coastal regions.
We present a new strongform meshless solver combined with the boundary conditionenforced immersed boundary method for the numerical solution of the nonstationary, incompressible, viscous Navier–Stokes equations in their stream functionvorticity (in 2D) and vector potentialvorticity (in 3D) formulation. We use a Cartesian grid to discretize the spatial domain. We apply explicit time integration to update the transient vorticity equations. Spatial derivatives of the unknown field functions are computed using the discretizationcorrected particle strength exchange method.
We present a strong form meshless solver for numerical solution of the nonstationary, incompressible, viscous Navier–Stokes equations in two (2D) and three dimensions (3D). We solve the flow equations in their stream functionvorticity (in 2D) and vector potentialvorticity (in 3D) formulation, by extending to 3D flows the boundary conditionenforced immersed boundary method, originally introduced in the literature for 2D problems. We use a Cartesian grid, uniform or locally refined, to discretize the spatial domain. We apply an explicit time integration scheme to update the transient vorticity equations, and we solve the Poisson type equation for the stream function or vector potential field using the meshless point collocation method. Spatial derivatives of the unknown field functions are computed using the discretizationcorrected particle strength exchange method. We verify the accuracy of the proposed numerical scheme through commonly used benchmark and example problems. Excellent agreement with the data from the literature was achieved. The proposed method was shown to be very efficient, having relatively large critical time steps.
The findings obtained as a result of the study showed that artificial neural networks are an ideal tool that can be used to model Darcy–Forchheimer Ree–Eyring fluid flow towards a permeable stretch layer with activation energy and a convective boundary condition.
In this study, Darcy Forchheimer flow paradigm, which is a useful paradigm in fields such as petroleum engineering where high flow velocity effects are common, has been analyzed with artificial intelligence approach. In this context, first of all, Darcy–Forchheimer flow of Ree–Eyring fluid along a permeable stretching surface with convective boundary conditions has been examined and heat and mass transfer mechanisms have been investigated by including the effect of chemical process, heat generation/absorption, and activation energy. Cattaneo–Christov heat flux model has been used to analyze heat transfer properties. Within the scope of optimizing Darcy–Forchheimer flow of Ree–Eyring fluid; three different artificial neural network models have been developed to predict Nusselt number, Sherwood number, and skin friction coefficient values. The developed artificial neural network model has been able to predict Nusselt number, Sherwood number, and skin friction coefficient values with high accuracy. The findings obtained as a result of the study showed that artificial neural networks are an ideal tool that can be used to model Darcy–Forchheimer Ree–Eyring fluid flow towards a permeable stretch layer with activation energy and a convective boundary condition.
With the exception of thin rigid bodies, the DFIB approach successfully models FSI problems. Modeling thin rigid structures is a significant challenge when employing the DFIB approach to generate FSI solutions. A VOSbased algorithm with DFIB was developed to model thin and volumeless rigid bodies.
The new capability has been added as the numerical method for modeling volumeless and thin rigid bodies to the direct forcing immersed boundary (DFIB) method. The DFIB approach is based on adding a virtual force to the Navier–Stokes equations of incompressible flow to account for the interaction between the fluid and structures. The volume of a solid function (VOS) identifies the stationary or moving solid structures in a given fluid domain. A new VOSbased algorithm was developed to identify thin, rigid structure boundary points in fluid flow and ensure that the fluid cannot cross through the boundary of a thin rigid structure while moving or stationary. The DFIB method was first validated in a threedimensional (3D) turbulent flow over a circular cylinder. The largeeddy simulation simulated the turbulent flow scales. The proposed algorithm was tested using a 3D turbulent flow past a stationary and rotating Savonius wind turbine that functions as a thin, rigid body. The validation results showed that the selected DFIB approach, combined with the novel algorithm, could simulate a thin, volumeless, rigid structure that is stationary and rotating in incompressible turbulent flows. The current method is also applicable for twoway fluidstructure interaction problems.
Publication date: 1 December 2022
Source: Journal of Computational Physics, Volume 470
Author(s): Dexuan Xie
Publication date: 1 December 2022
Source: Journal of Computational Physics, Volume 470
Author(s): Wenzhong Zhang, Wei Cai
Publication date: 1 December 2022
Source: Journal of Computational Physics, Volume 470
Author(s): Tianpei Cheng, Haijian Yang, Shuyu Sun
Publication date: 1 December 2022
Source: Journal of Computational Physics, Volume 470
Author(s): Firas Dhaouadi, Michael Dumbser
Publication date: Available online 27 September 2022
Source: Journal of Computational Physics
Author(s): Michel Bergmann, Antoine Fondanèche, Angelo Iollo
Publication date: Available online 26 September 2022
Source: Journal of Computational Physics
Author(s): Jiacheng Xu, Dan Hu, Han Zhou
Publication date: Available online 26 September 2022
Source: Journal of Computational Physics
Author(s): Junxiang Yang, Junseok Kim
Publication date: Available online 26 September 2022
Source: Journal of Computational Physics
Author(s): Ruo Li, Yixiao Lu, Yanli Wang, Haoxuan Xu
Publication date: Available online 26 September 2022
Source: Journal of Computational Physics
Author(s): Giuseppe Orlando, Paolo Francesco Barbante, Luca Bonaventura
Publication date: 1 December 2022
Source: Journal of Computational Physics, Volume 470
Author(s): D. Khimin, M.C. Steinbach, T. Wick
Determining the behavior of the leadingedge suction force, represented nondimensionally by the leadingedge suction parameter (LESP), can reliably help indicate the state of flow over the airfoil and therefore the force and moment characteristics. The current work aims at studying the variations in the LESP, forces, and pitching moment with freestream Reynolds number and airfoil thickness in unsteady flows. Computational data for the NACA 0012, 0015, and 0018 airfoils undergoing a baseline pitching motion over a range of freestream Reynolds number conditions are analyzed. The critical LESP, which is the instantaneous value of LESP at leadingedge vortex initiation, is observed to first decrease and subsequently increase with Reynolds number. This behavior can be correlated to the rate at which leadingedge flow curvature increases with Reynolds number. Thicker airfoils are observed to sustain a larger amount of suction force prior to breakdown and ensuing leadingedge vortex (LEV) shedding. Lift, drag, and moment are found to be dependent on thickness and Reynolds number prior to LEV shedding due to differences in the boundary layer characteristics, but independent after suction breakdown due to the similarity in LEV dynamics. These findings serve to support the development of a more generalized definition of a suctionforce parameter that is independent of flow conditions and airfoil geometry.
Direct numerical simulation and theoretical analysis of acoustic receptivity are performed for the boundary layer on a flat plate in Mach 6 flow at various angles of attack (AoA). Slow or fast acoustic wave passes through: a bow shock at AoA \(=5^{\circ }\) , a weak shock induced by the viscous–inviscid interaction at AoA \(=0^{\circ }\) or an expansion fan emanating from the plate leading edge at AoA \(=5^{\circ }\) . The study is focused on cases where the integral amplification of unstable mode S (or Mack second mode) is sufficiently large \((N\approx 8.4)\) to be relevant to transition in lowdisturbance environments. It is shown that excitation of dominant modes F and S occurs in a small vicinity of the plate leading edge. The initial disturbance propagates further downstream in accord with the twomode approximation model accounting for the meanflow nonparallel effects and the intermodal exchange mechanism. This computationally economical model can be useful for predictions of the second mode dominated transition onset using the physicsbased amplitude method.
In the case of microscopic particles, the momentum exchange between the particle and the gas flow starts to deviate from the standard macroscopic particle case, i.e. the noslip case, with slip flow occurring in the case of low to moderate particle Knudsen numbers. In order to derive new drag force models that are valid also in the slip flow regime for the case of nonspherical particles of arbitrary shapes using computational fluid dynamics, the noslip conditions at the particle surface have to be modified in order to account for the velocity slip at the surface, mostly in the form of the Maxwell’s slip model. To allow a continuous transition in the boundary condition at the wall from the noslip case to the slip cases for various Knudsen (Kn) number value flow regimes, a novel specific slip length model for the use with the Maxwell boundary conditions is proposed. The model is derived based on the data from the published experimental studies on spherical microparticle drag force correlations and Cunninghambased slip correction factors at standard conditions and uses a detailed CFD study on microparticle fluid dynamics to determine the correct values of the specific slip length at selected Kn number conditions. The obtained data on specific slip length are correlated using a polynomial function, resulting in the specific slip length model for the noslip and slip flow regimes that can be applied to arbitrary convex particle shapes.
The problem of a solitary surface gravity wave in a flow of an inviscid incompressible fluid in a channel of constant depth is considered. The problem is solved in twodimensional formulation. The wave moves at a constant speed. In a coordinate system moving along with the wave, the flow is stationary. Its mathematical model is reduced to a boundary value problem for a strip in the complex potential plane. This is converted to a boundary value problem for a halfplane by conformal mapping. The solution is obtained using a Cauchytype integral for the density of which a nonlinear integral equation is derived. Its solution is found with the Galerkin method and the Newton–Raphson technique. The calculated results are compared with the experimental data and the calculations by other researchers. The lower limit of the speed of a solitary wave is found. The advantage of the proposed method is the simplicity of the resulting integral equation, which makes it possible to effectively apply numerical methods of solution.
This study investigates the stability of compressible swirling wake flows including the viscous effects using linear stability theory. A spatial stability analysis is performed to evaluate the influence of the axial velocity deficit and circulation as well as the Reynolds number and Mach number as the main parameters that affect the instability. The growth rates of the unstable modes at several azimuthal wavenumbers are compared. The maximum growth rates and their dependency with respect to each parameter are analyzed. It is confirmed that the instability monotonically increases as the axial velocity deficit increases. For small axial velocity deficit, characteristics that are different from the results reported using inviscid analysis are identified and analyzed. Additionally, a decrease in instability is observed as the viscous and compressibility effects become stronger. In terms of circulation, it is confirmed that there is a certain region of circulation that exhibits maximum instability. The stability analysis is expected to serve as a part of a useful methodology for preliminary design and parametric study for engineering problems such as vortex generators in highspeed flows, owing to both efficiency and accuracy.
A volume of fluid method combined with an adaptive grid method was used to study the influence of Galilei (Ga) and Eötvös (Eo) numbers and characteristic parameters (such as rheological index (n) and characteristic time ( \(\lambda \) )) of shearthinning liquids on the hydrodynamics of two types of unsteady bubbles. One is the bubble with central breakup behaviors, of which the rise trajectory is a straight line and the shape is symmetrical; however, the shape and centroid velocity cannot reach a steady state. Bubble shape becomes annular after radial expansion, and the centroid velocity has two peaks. The other is the unsteady bubble, of which the rise trajectory is zigzag, but both the shape and rise velocity cannot reach a steady state. The shape of this unstable bubble is flat, which causes periodic vortex shedding at the tail of a bubble. Thus, bubble rise velocity cannot reach a steady state. When the influence of viscous force is relatively weak and Eo is in the range of 50–55, a bubble shows central breakup behaviors. When Eo is low (Eo \(<10\) ), effective Morton numbers (Mo \(^{\mathrm{eff}}\) ) decrease to the magnitude of \(10^{7}\) and effective Reynolds numbers meet the condition of Re \(^{\mathrm{eff}}\ge 125.2\) , a bubble shows the second type of unsteady characteristics.
The use of multitaper estimates for spectral proper orthogonal decomposition (SPOD) is explored. Multitaper and multitaperWelch estimators that use discrete prolate spheroidal sequences (DPSS) as orthogonal data windows are compared to the standard SPOD algorithm that exclusively relies on weighted overlapped segment averaging, or Welch’s method, to estimate the crossspectral density matrix. Two sets of turbulent flow data, one experimental and the other numerical, are used to discuss the choice of resolution bandwidth and the biasvariance tradeoff. MultitaperWelch estimators that combine both approaches by applying orthogonal tapers to overlapping segments allow for flexible control of resolution, variance, and bias. At additional computational cost but for the same data, multitaperWelch estimators provide lower variance estimates at fixed frequency resolution or higher frequency resolution at similar variance compared to the standard algorithm.
Motivated by recent advances in the development of the numerical calculation of fine flow in liquid film, the thermocapillary convection in thin liquid film (1mm) due to temperature difference is studied in this paper. To describe the formation of the thermocapillary convection on gasliquid interface, a twophase system was designed, in which the momentum and energy interact directly through the free surface. The finite volume method is used to solve the NS equation in gas phase and liquid phase, respectively, and the velocity and temperature information are exchanged on the free surface in each time step. The results show that a thermocapillary wave appears in the liquid film when the temperature difference exceeds a certain value. The temperature and velocity fluctuations on the free surface show a radiation shape. The flow field structure is completely symmetrical in the basic state, but it is axisymmetric in the case of oscillation state. The propagation direction of thermocapillary wave is affected by many factors (ambient temperature or inner wall rotation). The wave propagation direction is consistent with the rotation direction when the inner wall rotates. When the angular velocity of inner wall rotation is 8 rad/s, the wave number of thermocapillary wave will be reduced to 3, which is independent of the rotation direction.
The influence of turbulence inflow generation on direct numerical simulations (DNS) of highspeed turbulent boundary layers at Mach numbers of 2 and 5.84 is investigated. Two main classes of inflow conditions are considered, based on the recycling/rescaling (RR) and the digital filtering (DF) approach, along with suitably modified versions. A series of DNS using very long streamwise domains is first carried out to provide reliable data for the subsequent investigation. A set of diagnostic parameters is then selected to verify achievement of an equilibrium state, and correlation laws for those quantities are obtained based on benchmark cases. Simulations using shorter domains, with extent comparable with that used in the current literature, are then carried out and compared with the benchmark data. Significant deviations from equilibrium conditions are found, to a different extent for the various flow properties, and depending on the inflow turbulence seeding. We find that the RR method yields superior performance in the evaluation of the innerscaled wall pressure fluctuations and the turbulent shear stress. DF methods instead yield quicker adjustment and better accuracy in the prediction of wall friction and of the streamwise Reynolds stress in supersonic cases. Unrealistically high values of the wall pressure variance are obtained by the baseline DF method, while the proposed DF alternatives recover a closer agreement with respect to the benchmark. The hypersonic test case highlights that similar distribution of wall friction and heat transfer are obtained by both RR and DF baseline methods.
The problem of the linear stability of the stratified Kolmogorov flow driven by a sinusoidal in space force in a viscous and diffusive Boussinesq fluid is revisited using the Floquet theory, Galerkin approximations and the method of (generalized) continued fractions. Numerical and analytical arguments are provided in favor of a conjecture that an ideal stratified Kolmogorov flow is prone to shortwave instability for Richardson numbers markedly greater than the critical Richardson number Ri \(=\) ¼ that appears in the Miles–Howard theorem. The shortwave instability of the stratified Kolmogorov flow is conjectured to be due to a resonance amplification of the Dopplershifted internal gravity wave modes, in the presence of critical levels of the main flow that are ignored in the proof of the Miles–Howard theorem, but it is emphasized that the complete resolution of the above paradox is a task for future research.