Exponential boundary-layer approximation space for solving the compressible laminar Navier-Stokes equations

A new discontinuous Galerkin finite element method for solving diffusion problems is introduced. Unlike the traditional local discontinuous Galerkin method, the scheme called the direct discontinuous Galerkin (DDG) method is based on the direct weak formulation for solutions of parabolic equations in each computational cell and lets cells communicate via the numerical flux $\widehat{u_x}$ only. We propose a general numerical flux formula for the solution derivative, which is consistent and conservative; and we then introduce a concept of admissibility to identify a class of numerical fluxes so that the nonlinear stability for both one-dimensional (1D) and multidimensional problems are ensured. Furthermore, when applying the DDG scheme with admissible numerical flux to the 1D linear case, kth order accuracy in an energy norm is proven when using kth degree polynomials. The DDG method has the advantage of easier formulation and implementation and efficient computation of the solution. A series of numerical examples are presented to demonstrate the high order accuracy of the method. In particular, we study the numerical performance of the scheme with different admissible numerical fluxes.

The direct discontinuous Galerkin (DDG) methods on a Bakhvalov‐type mesh are developed to solve a singularly perturbed reaction‐diffusion problem. First, we introduce and analyze a unified, generalized framework for the DDG methods. Then, the parameter‐uniform convergence rate of our numerical method in the balanced norm is established by using the local weighted ‐projection, where represents the degree of the piecewise polynomial space. Finally, some numerical results are presented to further validate our theoretical findings.

The direct discontinuous Galerkin (DDG) viscous flux scheme for the high order spectral volume method

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for one‐dimensional linear convection‐diffusion equations. We prove, under some suitable choice of numerical fluxes and initial discretization, a 2k‐th and ‐th order superconvergence rate of the DDG approximation at nodes and Lobatto points, respectively, and a ‐th order of the derivative approximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDG solution is superconvergent with an order k + 2 to a particular projection of the exact solution. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 290–317, 2017

We provide a framework for the analysis of the direct discontinuous Galerkin (DDG) methods for multi-dimensional convection–diffusion equations subject to various boundary conditions. A key tool is the global projection constructed by the DDG scheme applied to an associated elliptic problem. Such projection is well-defined for a class of diffusive flux parameters, and the optimal projection error in $$L^2$$ is obtained with an arbitrary locally regular partition of the domain and for an arbitrary degree of polynomials. This results in the optimal $$L^2$$ error for the DDG method to the elliptic problem, and further leading to the optimal $$L^2$$ error for the DDG method to the convection–diffusion problem.

In this paper, we are concerned with the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity μ(ρ) = ρα(α > 0) and pressure P(ρ) = ργ (γ > 1). We will establish the global existence and asymptotic behavior of weak solutions for any α > 0 and γ > 1 under the assumption that the density function keeps a constant state at far fields. In particular, in the case that \documentclass[12pt]{minimal}\begin{document}$0<\alpha <\frac{1}{2}$\end{document}0<α<12, we obtain the large time behavior of the strong solution obtained by Mellet and Vasseur when the solution has a lower bound (no vacuum).

A new direct discontinuous Galerkin method with interface correction for two-dimensional compressible Navier-Stokes equations

This paper deals with the study of an incompressible electro-magneto-hydrodynamic (EMHD) Jeffrey fluid flow over a vertical nonlinear stretching surface of variable thickness. Heat and mass transfer effects are analyzed by considering different source terms like viscous dissipation, Ohmic heating, thermophoresis, Brownian motion, thermal heat source, exponential heat source and activation energy. Governing equations for the flow system are converted into dimensionless forms using appropriate similarity transformations. The solution for the resulting governing equations is obtained by using the shooting technique with RK-4 method. The effects of various physical parameters such as magnetic field parameter [Formula: see text], Grashof number (Gr), solutal Grashof number [Formula: see text], Brownian diffusion parameter [Formula: see text], thermophoresis diffusion parameter [Formula: see text], thermal heat source parameter [Formula: see text], exponential heat source parameter [Formula: see text], Prandtl number (Pr) and Lewis number (Le) are presented with the help of graphs. It is observed that the heat transfer effects increase by increasing thermal and exponential heat sources, and mass transfer effects enhance by increasing the activation energy. Entropy generation for this flow system is also analyzed. Entropy decreases with an increase in the electric field parameter. In contrast, the Bejan number initially increases with an increase in the electric field parameter. After some particular value of electric field parameter, it changes its behavior in the boundary layer and decreases with an increase in the electric field parameter. Entropy and Bejan number increase with an increment in the concentration difference parameter. The accuracy of the results is validated by those of published literature and found in reasonable justification. The present results may be helpful in many engineering and industrial applications like manufacturing lubrication, natural gas networks, cooling nuclear reactors and spray processes.

We are interested in front propagation problems in the presence of obstacles. We extend a previous work (Bokanowski et al. SIAM J Sci Comput 33(2):923–938, 2011), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of Bokanowski et al. (SIAM J Control Optim 48(7):4292–4316, (2010)), leading to a level set formulation driven by $$\min (u_t + H(x,\nabla u), u-g(x))=0$$ , where $$g(x)$$ is an obstacle function. The DG scheme is motivated by the variational formulation when the Hamiltonian $$H$$ is a linear function of $$\nabla u$$ , corresponding to linear convection problems in the presence of obstacles. The scheme is then generalized to nonlinear equations, written in an explicit form. Stability analysis is performed for the linear case with Euler forward, a Heun scheme and a Runge-Kutta third order time discretization using the technique proposed in Zhang and Shu (SIAM J Numer Anal 48:1038–1063, 2010). Several numerical examples are provided to demonstrate the robustness of the method. Finally, a narrow band approach is considered in order to reduce the computational cost.

Multi-cycle direct numerical simulations (DNS) of a laboratory-scale engine at technically relevant engine speeds (1500 and 2500 rpm) are performed to investigate the transient velocity and thermal boundary layers (BL) as well as the wall heat flux during the compression stroke under motored operation. The time-varying wall-bounded flow is characterized by a large-scale tumble vortex, which generates vortical structures as the flow rolls off the cylinder wall. The bulk flow is found to strongly affect the development of the BL profiles, especially at higher engine speeds. As a result, the large-scale flow structures lead to alternating pressure gradients near the wall, invalidating the flow equilibrium assumptions used in typical wall modeling approaches. The thickness of the velocity BL and of the viscous sublayer was found to scale inversely with engine speed and crank angle. The thermal BL thickness also scales inversely with engine speed but increases with in-cylinder temperature. In contrast, thermal displacement thickness, which is sometimes used as a proxy for thermal BL thickness, was found to decrease with increasing temperature in the bulk. Examination of the heat flux distribution revealed areas of increased heat flux, particularly at places characterized by strong flow directed towards the wall. In addition, significant cyclic variations in the surface-averaged wall heat flux were observed for both engine speeds. An analysis of the cyclic tumble ratio revealed that the cycles with lower tumble ratio values near top dead center (TDC), indicative of an earlier tumble breakdown, also exhibit higher surface averaged wall heat fluxes. These findings extend previous numerical and experimental results for the evolution of BL structure during the compression stroke and serve as an important step for future engine simulations under realistic operating conditions.

In this paper we present a set of wave flume experiments for a solitary wave reflecting off a vertical wall. A particle tracking velocimetry (PTV) technique is used to measure free-surface velocity and the velocity field in the vicinity of the moving contact line. We observe that the free surface undergoes the so-called rolling motion as the contact line moves up and down the vertical wall, and fluid particles on the free surface almost always flow toward the wall except at the end of the reflection process. As the contact line descends along the wall, wall boundary layer flows move in a downward direction and therefore the boundary layer acts like a conduit through which the surface-rolling-induced flow escapes from the meniscus. However, during the last phase of the reflection process flow reversal occurs inside the wall boundary layer. An approximate analytical solution is developed to explain the flow reversal feature. Very good agreement between the approximate theory and measured data is obtained. Because of the flow reversal, boundary layer flows collide with the surface-rolling-induced flows. The collision gives rise to a jet ejecting from the meniscus into the water body, which later evolves into a small eddy. It is noticed that the fluid particles in different regions such as the free stream, the free-surface boundary layer and the wall boundary layer, can be transported to other regions by passing through the meniscus.

Forced-convection heat transfer in a heated working fluid at a thermodynamic state near its pseudocritical point is poorly predicted by correlations calibrated with data at subcritical temperatures and pressures. This is suggested to be primarily due to the influence of large wall-normal thermophysical property gradients that develop in proximity of the pseudocritical point on the concentration of coherent turbulence structures near the wall. The physical mechanisms dominating this influence remain poorly understood. In the present study, direct numerical simulation is used to study the development of coherent vortical structures within a turbulent spot under the influence of large wall-normal property gradients. A turbulent spot rather than a fully turbulent boundary layer is used for the study, for the coherent structures of turbulence in a spot tend to be in a more organized state which may allow for more effective identification of cause-and-effect relationships. Large wall-normal gradients in thermophysical properties are created by heating the working fluid which is near the pseudocritical thermodynamic state. It is found that during improved heat transfer, wall-normal gradients in density accelerate the growth of the Kelvin-Helmholtz instability mechanism in the shear layer enveloping low-speed streaks, causing it to roll up into hairpin vortices at a faster rate. It is suggested that this occurs by the baroclinic vorticity generation mechanism which accelerates the streamwise grouping of vorticity during shear layer roll-up. The increased roll-up frequency leads to reduced streamwise spacing between hairpin vortices in wave packets. The density gradients also promote the sinuous instability mode in low-speed streaks. The resulting oscillations in the streaks in the streamwise-spanwise plane lead to locally reduced spanwise spacing between hairpin vortices forming over adjacent low-speed streaks. The reduction in streamwise and spanwise spacing between hairpin vortices causes them to interact more frequently by merging together and by breaking apart into smaller turbulence structures.

In this paper, we consider the free boundary value problem (FBVP) for one-dimensional isentropic compressible Navier-Stokes equations (CNS) with density-dependent viscosity coefficient and constant exterior pressure. Under certain assumptions imposed on the initial data, the global existence and uniqueness of a strong solution to FBVP for CNS are established, in particular, the strong solution tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity.

Comparison of numerical methods for the solution of viscous incompressible and low Mach number compressible flow

Energy integral equation for premixed flame-wall interaction in turbulent boundary layers and its application to turbulent burning velocity and wall flux evaluations