Viscous Case Research Articles

Three dimensional shock wave/boundary layer interactions

An investigation into a three-dimensional, curved shock wave interacting with a three-dimensional, curved boundary layer on a slender body is presented. Three different nose profiles mounted on a cylindrical body were tested in a supersonic wind tunnel and numerically simulated by solving the Navier–Stokes equations. The conical and hemispherical nose profiles tested were found to generate shock waves of sufficient strength to separate the boundary layer on the cylinder, while the shock wave generated by the ogival profile did not separate the boundary layer. For the separated flow, separation was found to occur predominantly on the windward side of the cylinder with the lee-side remaining shielded from the direct impact of the incident shock wave. A thickening of the boundary layer on the lee-side of all the profiles was observed, and in the conical and hemispherical cases this leads to the re-formation of the incident shock wave some distance away from the surface of the cylinder. A complex reflection pattern off the shock wave/boundary layer interaction (SWBLI) was also identified for the separated flow cases. For comparative purposes, an inviscid simulation was performed using the hemispherical profile. Significant differences between the viscous and inviscid results were noted including the absence of a boundary layer leading to a simplified shock wave reflection pattern forming. The behaviour of the incident shock wave on the lee-side of the cylinder was also affected with the shock wave amalgamating on the surface of the cylinder instead of away from the surface as per the viscous case. Test data from the wind tunnel identified two separation lines present on the cylindrical surface of the hemispherical SWBLI generator. The pair of lines were not explicitly evident in the original CFD simulations run, but were later identified in a high-resolution simulation.

Waves in two-layer shear flow for viscous and inviscid fluids

A two-layer shear flow is studied for inviscid and viscous fluids. Here, the layers flow between two horizontal walls and are buoyantly stable. Each layer contains a finite amount of shear and the horizontal velocity is specified such that it is continuous when unperturbed. The interface between the two layers is given a small sinusoidal perturbation and the subsequent response of the system is studied. Different solution techniques are employed for the inviscid and viscous flows. These both rely on linearizing the governing equations for each of these flows. In particular, the viscous flow is constrained to remain within a small perturbation of the unperturbed flow as it evolves. This assumption is justified since standing wave behaviour is expected in the inviscid case. Solutions are presented for a variety of different values of the shear parameters and the way these parameter choices affect the interaction between vorticity and density in the viscous case is investigated in detail. These linearized solutions are confirmed by comparison with fully non-linear results obtained numerically.

Stirring by squirmers

We analyse a simple ‘Stokesian squirmer’ model for the enhanced mixing due to swimming micro-organisms. The model is based on a calculation of Thiffeault & Childress (Phys. Lett. A, vol. 374, 2010, p. 3487), where fluid particle displacements due to inviscid swimmers are added to produce an effective diffusivity. Here we show that, for the viscous case, the swimmers cannot be assumed to swim an infinite distance, even though their total mass displacement is finite. Instead, the largest contributions to particle displacement, and hence to mixing, arise from random changes of direction of swimming and are dominated by the far-field stresslet term in our simple model. We validate the results by numerical simulation. We also calculate non-zero Reynolds number corrections to the effective diffusivity. Finally, we show that displacements due to randomly swimming squirmers exhibit probability distribution functions with exponential tails and a short-time superdiffusive regime, as found previously by several authors. In our case, the exponential tails are due to ‘sticking’ near the stagnation points on the squirmer's surface.

Comparative study of different numerical approaches in space–time CESE framework for high-fidelity flow simulations

With the advancement of computer hardware, the trend of research in computational fluid dynamics is moving towards development of highly accurate, unstructured-mesh compatible, robust and efficient numerical methods for simulating problems involving strong transient effects and relatively complex geometries as well as physics. The space–time conservation element and solution element method is a genuinely multi-dimensional, unstructured-mesh compatible numerical framework, which was built from a consistent and synergetic integration of conservation laws in the space–time domain to avoid the limitations of conventional schemes, such as the use of 1-D flux reconstruction with a Riemann solver. It has been shown that the framework can be used for time-accurate simulations of a variety of problems involving unsteady waves, strong flow discontinuities, and their interactions with remarkable accuracy. However, this method at its current state has encountered the challenge in balancing the robustness and numerical accuracy when highly stretched meshes were used in viscous flow simulation. In this paper, we briefly discuss various numerical approaches developed for this framework thus far as well as their strengths and weaknesses, and conduct a comparative study of their numerical accuracies using some 2-D viscous benchmark test cases. The application of this method in realistic, complex 3-D problems is also included here to demonstrate its computational efficiency in large-scale computing.

Serrin-Type Criterion for the Three-Dimensional Viscous Compressible Flows

We extend the well-known Serrin's blowup criterion for the three-dimensional (3D) incompressible Navier–Stokes equations to the 3D viscous compressible cases. It is shown that for the Cauchy problem of the 3D compressible Navier–Stokes equations in the whole space, the strong or smooth solution exists globally if the velocity satisfies the Serrin's condition and either the supernorm of the density or the $L^1(0,T;L^\infty)$-norm of the divergence of the velocity is bounded. Furthermore, in the case that either the shear viscosity coefficient is suitably large or there is no vacuum, the Serrin's condition on the velocity can be removed in this criterion.

High resolution finite volume computations on unstructured grids using solution dependent weighted least squares gradients

An improved high resolution finite volume method based on linear and quadratic variable reconstructions using solution dependent weighted least squares (SDWLS) gradients has been presented here. An extended stencil consisting of vertex-based neighbours of a cell is used in the higher order reconstructions for inviscid flux computations. A QR algorithm with Householder transformation is used to solve the weighted least squares problem. In case of Navier–Stokes equations, viscous fluxes are discretized in a central differencing manner based on the Coirier’s diamond path. A few inviscid and viscous test cases are solved in order to demonstrate the efficacy of the present method. Progressive improvements in solution accuracy are observed with the increase in the order of variable reconstructions. In most cases, results of quadratic reconstruction show significant improvements over that of linear reconstruction.

Nonlinear and viscous effects on wave propagation in an elastic axisymmetric vessel

In this paper, a power series and Fourier series approach is used to solve the governing equations of motion in an elastic axisymmetric vessel with the assumption that the fluid is incompressible and Newtonian in a laminar flow. We obtain solutions for the wave speed and attenuation coefficient, analytically where possible, and show how these differ under a number of different conditions. Viscosity is found to reduce the wave speed from that predicted by linear wave theory and the nonlinear terms to increase the wave speed in comparison to the linear solution. For vessels with a wall stiffness in the arterial range, the reduction in the wave speed due to the viscous terms is approximately 10% and the increase due to the nonlinear terms is approximately 5%. This difference between the linear and nonlinear wave speeds was found to be largely constant irrespective of the number of terms considered in the power series for the velocity profile. The linear wave speed was found to vary weakly with stiffness, whilst the nonlinear wave speed was found to vary significantly with the stiffness, especially at low values of stiffness. The 10% variation in the wave speed due to the viscous terms was found to be constant with wall stiffness whilst the 5% variation due to the nonlinear terms was found to vary with wall stiffness. The importance of the number of terms considered in the power series is discussed showing that only a relatively small number is required in the viscous case to obtain accurate results.

Hubble parameter in QCD Universe for finite bulk viscosity

AbstractWe consider the influence of the perturbative bulk viscosity on the evolution of the Hubble parameter in the QCD era of the early Universe. For the geometry of the Universe we assume the homogeneous and isotropic Friedmann‐Lemaitre‐Robertson‐Walker metric, while the background matter is assumed to be characterized by barotropic equations of state, obtained from recent lattice QCD simulations, and heavy‐ion collisions, respectively. Taking into account a perturbative form for the bulk viscosity coefficient, we obtain the evolution of the Hubble parameter, and we compare it with its evolution for an ideal (non‐viscous) cosmological matter. A numerical solution for the viscous QCD plasma in the framework of the causal Israel‐Stewart thermodynamics is also obtained. Both the perturbative approach and the numerical solution qualitatively agree in reproducing the viscous corrections to the Hubble parameter, which in the viscous case turns out to be slightly different as compared to the non‐viscous case. Our results are strictly limited within a very narrow temperature‐ or time‐interval in the QCD era, where the quark‐gluon plasma is likely dominant.

Spectral p‐multigrid discontinuous Galerkin solution of the Navier–Stokes equations

AbstractDiscontinuous Galerkin (DG) methods are very well suited for the construction of very high‐order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods, a high‐order spectral element DG approximation of the Navier–Stokes equations coupled with a p‐multigrid solution strategy based on a semi‐implicit Runge–Kutta smoother is considered here. The effectiveness of the proposed approach in the solution of compressible shockless flow problems is demonstrated on 2D inviscid and viscous test cases by comparison with both a p‐multigrid scheme with non‐spectral elements and a spectral element DG approach with an implicit time integration scheme. Copyright © 2010 John Wiley & Sons, Ltd.

Casimir effects near the big rip singularity in viscous cosmology

Analytical properties of the scalar expansion in the cosmic fluid are investigated, especially near the future singularity, when the fluid possesses a constant bulk viscosity \zeta. In addition, we assume that there is a Casimir-induced term in the fluid's energy-momentum tensor, in such a way that the Casimir contributions to the energy density and pressure are both proportional to 1/a^4, 'a' being the scale factor. A series expansion is worked out for the scalar expansion under the condition that the Casimir influence is small. Close to the Big Rip singularity the Casimir term has however to fade away and we obtain the same singular behavior for the scalar expansion, the scale factor, and the energy density, as in the Casimir-free viscous case.

Frequency response of cantilever beams immersed in compressible fluids with applications to the atomic force microscope

The dynamics of microcantilever beams can be strongly affected by immersion in fluid. While the importance of viscosity for devices of microscale dimensions is well established, the significance of fluid compressibility has not been investigated in detail. Here, we present a rigorous theoretical model for the frequency response of a rectangular cantilever beam that is executing normal and torsional oscillations, and is immersed in a compressible fluid. Both the viscous case and the inviscid limit are considered, and the model is valid for arbitrary mode number. We find that compressibility becomes increasingly important as the mode number rises. This is particularly relevant for gases, where compressibility is found to be important for high mode numbers of practical interest.

Stability of global and exponential attractors for a three‐dimensional conserved phase‐field system with memory

AbstractWe consider a conserved phase‐field system on a tri‐dimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ϑ, which is represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore, the system consists of a linear integrodifferential equation for ϑ, which is coupled with a viscous Cahn–Hilliard type equation governing the order parameter χ. The latter equation contains a nonmonotone nonlinearity ϕ and the viscosity effects are taken into account by a term −αΔ∂tχ, for some α⩾0. Rescaling the kernel k with a relaxation time ε>0, we formulate a Cauchy–Neumann problem depending on ε and α. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {ℰα,ε} for our problem, whose basin of attraction can be extended to the whole phase–space in the viscous case (i.e. when α>0). Moreover, we prove that the symmetric Hausdorff distance of ℰα,ε from a proper lifting of ℰα,0 tends to 0 in an explicitly controlled way, for any fixed α⩾0. In addition, the upper semicontinuity of the family of global attractors {𝒜α,ε} as ε→0 is achieved for any fixed α>0. Copyright © 2009 John Wiley & Sons, Ltd.

Effects of viscosity ratio on deformation of a viscoelastic drop in a Newtonian matrix under steady shear

Deformation of an Oldroyd B drop in a Newtonian matrix under steady shear is simulated using a front tracking finite difference method for varying viscosity ratio. For drop viscosity lower than that of the matrix, the long-time steady deformation behavior is similar to that of the viscosity matched system—the drop shows reduced deformation with increasing Deborah number due to the increased inhibiting viscoelastic normal stress inside the drop. However for higher viscosity ratio systems, the drop response is non-monotonic—the steady drop deformation first decreases with increasing Deborah number but above a critical Deborah number, it increases with further increase in Deborah number, reaching higher than the viscous case value for some viscosity ratios. We explain the increase in deformation with Deborah number by noting that at higher viscosity ratios, strain rate inside the drop is reduced, thereby reducing the inhibiting viscoelastic stress. Furthermore, similar to the viscosity matched system, the drop inclination angle increases with increasing Deborah number. A drop aligned more with the maximum stretching axis at 45 degree of the imposed shear, experiences increased viscous stretching. With increased ratio of polymeric viscosity to total drop viscosity, the drop deformation decreases and the inclination angle increases. Our simulation results compare favorably with a number of experimental and computational results from other researchers.

The trajectory and stability of a spiralling liquid jet: Viscous theory

We examine a spiralling slender viscous jet emerging from a rapidly rotating orifice, extending Wallwork et al. [I.M. Wallwork, S.P. Decent, A.C. King, R.M.S.M. Schulkes, The trajectory and stability of a spiralling liquid jet. Part 1. Inviscid theory, J. Fluid Mech. 459 (2002) 43–65] by incorporating viscosity. The effects of viscosity on the trajectory of the jet and its linear instability are determined using a mixture of computational and asymptotic methods, and verified using experiments. A non-monotonic relationship between break-up length and rotation rate is demonstrated with the trend varying with viscosity. The sizes of the droplets produced by this instability are determined by considering the most unstable wave mode. It is also found that there is a non-monotonic relationship between droplet size and viscosity. Satellite droplet formation is also considered by analysing very short wavelength modes. The effects of long wavelength modes are examined, and a wave which propagates down the trajectory of the jet is identified for the highly viscous case. A comparison between theoretical and experimental results is made, with favourable agreement. In particular, a quantitative comparison is made between droplet sizes predicted from the theory with experimental observations, with encouraging agreement obtained. Four different types of break-up are identified in our experiments. The experimentally observed break-up mechanisms are discussed in light of our theory.

Preconditioning methods for discontinuous Galerkin solutions of the Navier–Stokes equations

A Newton–Krylov method is developed for the solution of the steady compressible Navier–Stokes equations using a discontinuous Galerkin (DG) discretization on unstructured meshes. Steady-state solutions are obtained using a Newton–Krylov approach where the linear system at each iteration is solved using a restarted GMRES algorithm. Several different preconditioners are examined to achieve fast convergence of the GMRES algorithm. An element Line-Jacobi preconditioner is presented which solves a block-tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (Block-ILU(0)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling. This reordering is shown to be superior to standard reordering techniques (Nested Dissection, One-way Dissection, Quotient Minimum Degree, Reverse Cuthill–Mckee) especially for viscous test cases. The Block-ILU(0) factorization is performed in-place and an algorithm is presented for the application of the linearization which reduces both the memory and CPU time over the traditional dual matrix storage format. Additionally, a linear p-multigrid preconditioner is also considered, where Block-Jacobi, Line-Jacobi and Block-ILU(0) are used as smoothers. The linear multigrid preconditioner is shown to significantly improve convergence in term of number of iterations and CPU time compared to a single-level Block-Jacobi or Line-Jacobi preconditioner. Similarly the linear multigrid preconditioner with Block-ILU smoothing is shown to reduce the number of linear iterations to achieve convergence over a single-level Block-ILU(0) preconditioner, though no appreciable improvement in CPU time is shown.

Comparison of advanced large-scale minimization algorithms for the solution of inverse ill-posed problems

We compare the performance of several robust large-scale minimization algorithms for the unconstrained minimization of an ill-posed inverse problem. The parabolized Navier–Stokes equation model was used for adjoint parameter estimation. The methods compared consist of three versions of nonlinear conjugate-gradient (CG) method, quasi-Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS), the limited-memory quasi-Newton (L-BFGS) [D.C. Liu and J. Nocedal, On the limited memory BFGS method for large scale minimization, Math. Program. 45 (1989), pp. 503–528], truncated Newton (T-N) method [S.G. Nash, Preconditioning of truncated Newton methods, SIAM J. Sci. Stat. Comput. 6 (1985), pp. 599–616, S.G. Nash, Newton-type minimization via the Lanczos method, SIAM J. Numer. Anal. 21 (1984), pp. 770–788] and a new hybrid algorithm proposed by Morales and Nocedal [J.L. Morales and J. Nocedal, Enriched methods for large-scale unconstrained optimization, Comput. Optim. Appl. 21 (2002), pp. 143–154]. For all the methods employed and tested, the gradient of the cost function is obtained via an adjoint method. A detailed description of the algorithmic form of minimization algorithms employed in the minimization comparison is provided. For the inviscid case, the CG-descent method of Hager [W.W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and efficient line search, SIAM J. Optim. 16 (1) (2005), pp. 170–192] performed the best followed closely by the hybrid method [J.L. Morales and J. Nocedal, Enriched methods for large-scale unconstrained optimization, Comput. Optim. Appl. 21 (2002), pp. 143–154], while in the viscous case, the hybrid method emerged as the best performed followed by CG [D.F. Shanno and K.H. Phua, Remark on algorithm 500. Minimization of unconstrained multivariate functions, ACM Trans. Math. Softw. 6 (1980), pp. 618–622] and CG-descent [W.W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and efficient line search, SIAM J. Optim. 16 (1) (2005), pp. 170–192]. This required an adequate choice of parameters in the CG-descent method as well as controlling the number of L-BFGS and T-N iterations to be interlaced in the hybrid method.

On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media

We consider the derivation of the Khokhlov-Zabolotskaya-Kuznetzov (KZK) equation from the nonlinear isentropic Navier-Stokes and Euler systems. The KZK equation is a mathematical model that describes the nonlinear propagation of a finite-amplitude sound pulse in a thermo-viscous medium. The derivation of the KZK equation has to date been based on the paraxial approximation of small perturbations around a given state of the Navier-Stokes system. However, this method does not guarantee that the solution of the initial Navier-Stokes system can be reconstructed from the solution of the KZK equation. We introduce a corrector function in the derivation ansatz that allows one to validate the KZK-approximation. We also give the analysis of other types of derivation. We prove the validation of the KZK-approximation for the non-viscous case, as well as for the viscous nonlinear and linear cases. The results are obtained in Sobolev spaces for functions periodic in one of the space variables and with a mean value of zero. The existence of a unique regular solution of the isentropic Navier-Stokes system in the half space with boundary conditions that are both periodic and mean value zero in time is also obtained.

DYNAMICS OF VISCOUS DISSIPATIVE GRAVITATIONAL COLLAPSE: A FULL CAUSAL APPROACH

The Misner and Sharp approach to the study of gravitational collapse is extended to the viscous dissipative case in both the streaming out and the diffusion approximations. The dynamical equation is then coupled to causal transport equations for the heat flux, the shear, and the bulk viscosity, in the context of Israel–Stewart theory, without excluding the thermodynamics viscous/heat coupling coefficients. The result is compared with previous works where these later coefficients were neglected and viscosity variables were not assumed to satisfy causal transport equations. Prospective applications of this result to some astrophysical scenarios are discussed.

A New Mechanical Model for Particle Transport by Surface Waves and Applications

We present a study of the behavior of a ball under the influence of gravity on a platform. A propagating surface wave travels on the surface of the platform while the platform remains motionless. This is a modification of the classical bouncing ball problem and describes the transport of particles by surface waves. Phase and velocity maps cannot be expressed in the explicit form due to implicit formulations, and no formal analytical analyses is possible. Numerical analysis shows that the transition to chaos is produced via a period doubling routewhich is a common property for classical bouncers. These numerical analysis have been carried out for the conservative and for the viscous cases and also for elastic and for inelastic collisions. The bouncing process can be sensitive to the initial conditions and can be useful for control techniques which can dramatically increase the effectiveness of particle transport in practical applications. Finally, we also consider the mechanical model of a particle sliding on a surface which is also important because it has important physical implications such as the transportation of thin films in biomedical applications, among others.

Bounds on Front Speeds for Inviscid and Viscous G-equations

G-equations are well-known front propagation models in combustion and describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are HamiltonJacobi equations with convex (L type) but non-coercive Hamiltonians. We study front speeds of both inviscid and viscous G-equations in mean zero flows, and compare the qualitative speed properties with those of quadratically nonlinear Hamilton-Jacobi equations and KPP (Kolmogorov-Petrovsky-Piskunov) fronts (with minimal speed). For the inviscid case, we analyze a variational solution formula (control representation) by choosing suitable test functions. For the viscous case, we analyze traveling front equations which agree with the cell problem of homogenization. We found that viscosity can drastically alter the front speed growth law of G-equations. Without viscosity, front speed grows like O(A/ logA) in cellular flows of large amplitude A. With proper viscosity, the front speed grows no faster than O( √ logA). In contrast, the KPP front speed grows like O(A) in general cellular flows at any fixed viscosity. The L type nonlinearity appearing in the G-equation makes the key difference.