Multimodels for Incompressible Flows

The Navier—Stokes equations for incompressible fluids are coupled to models of reduced complexity, such as Oseen and Stokes, and the corresponding transmission conditions are investigated. A mathematical analysis of the corresponding problems is carried out. Numerical results obtained by finite elements and spectral elements are shown on several flow fields of physical interest.     

Decay in Time of Incompressible Flows

In this paper we consider the Cauchy problem for incompressible flows governed by the Navier-Stokes or MHD equations. We give a new proof for the time decay of the spatial L₂ L_2 norm of the solution, under the assumption that the solution of the heat equation with the same initial data decays. By first showing decay of the first derivatives of the solution, we avoid some technical difficulties of earlier proofs based on Fourier splitting.     

Conical and Quasi-Conical Incompressible Fluid Flows

The solution of the problem of fluid flow inside a cone with a small vertex angle is obtained in closed form. The conditions of occurrence of singular separation are considered within the framework of conical flow theory. A class of conical flows in which the vorticity is transported along streamlines of the potential velocity component is detected.Quasi-conical incompressible fluid flow, i.~e. a flow inside and outside an axisymmetric body with power-law generators is defined by analogy with supersonic compressible fluid flow. The conditions under which the effect of vorticity and swirling is significant are found as a result of an inspection analysis. An approximate solution of the problem of fluid flow inside a zero corner is found.A coordinate expansion representing a plane analog of conical flow is constructed in the neighborhood of the separation point of a creeping flow on a smooth surface.     

Wall-Distance-Free Turbulence Models Applied to Incompressible Flows

Abstract

A comparative study of low-Re wall-distance-free (WDF) turbulence models in incompressible flows is presented. The study includes the WDF k-? and the three-equation WDF k-?-γ as well as two k-? models which invoke the distance from the wall. The models are implemented in conjunction with a characteristics-based method and an implicit unfactored scheme. Comparison with direct-numerical-simulation data reveals that the WDF models provide much more accurate results for the dissipation rate, especially in the near wall region. A grid refinement study further reveals that the models which explicitly involve the distance from the wall cannot capture the correct turbulence dissipation rate behaviour in the near-wall region even on the finest grid, where grid-independent solution is achieved. However, the low-Re WDF models require slightly more iterations than the other k - ? models to converge. Results are presented for channel, flat plate and backward-facing step flows.     

Computations of Optimal Controls for Incompressible Flows

Abstract

This paper is concerned with numerical solutions of optimal control problems for unsteady, viscous, incompressible flows. In general, controls can be of the distributed type (external body force) or Dirichlet type 7lpar;e.g., boundary velocity). Here, wc only consider the former case, although most of what we present is also applicable to the latter. Two different optimization objectives and associated solution methodologies are described. One involves a global-in-time functional, the other a local-in-time functional. Which method is preferred depends on the specific application. Some test computational results are presented.     

Incompressible Viscous Flows in Borderline Besov Spaces

We establish two new estimates for a transport-diffusion equation. As an application we treat the problem of global persistence of the Besov regularity with , for the two-dimensional Navier–Stokes equations with uniform bounds on the viscosity. We provide also an inviscid global result.     

On an Inviscid Model for Incompressible Two-Phase Flows with Nonlocal Interaction

We consider a diffuse interface model which describes the motion of an ideal incompressible mixture of two immiscible fluids with nonlocal interaction in two-dimensional bounded domains. This model consists of the Euler equation coupled with a convective nonlocal Cahn-Hilliard equation. We establish the existence of globally defined weak solutions as well as well-posedness results for strong/classical solutions.     

Young Measures Generated by Ideal Incompressible Fluid Flows

In their seminal paper, DiPerna and Majda (Commun Math Phys 108(4):667–689, 1987) introduced the notion of a measure-valued solution for the incompressible Euler equations in order to capture complex phenomena present in limits of approximate solutions, such as persistence of oscillation and development of concentrations. Furthermore, they gave several explicit examples exhibiting such phenomena. In this paper we show that any measure-valued solution can be generated by a sequence of exact weak solutions. In particular this gives rise to a very large, arguably too large, set of weak solutions of the incompressible Euler equations.     

Incompressible Viscous Fluid Flows in a Thin Spherical Shell

Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier–Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier–Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically stable for all Reynolds numbers.      

Examples of Steady Axisymmetric Flows of an Ideal Incompressible Fluid

Fluid Dynamics - Three-dimensional (axisymmetric) flows of an ideal incompressible fluid are considered in multiply connected regions. In a coordinate system fitted to the velocity potential and...     

Linear Dynamics of Perturbations in Flows with Constant Horizontal Shear

The dynamics of perturbations in shallow water and incompressible stratified fluid flows with constant horizontal shear are described using the nonmodal analysis. It is shown that the shear flow perturbations can be divided into two classes on the basis of the potential vorticity: rapidly oscillating wave perturbations with zero potential vorticity and slow vortex perturbations with nonzero potential vorticity. In the cases of weak and strong shear the main features of the dynamics of wave and vortex perturbations are studied analytically (using the WKBG method) and numerically. It is shown that for large times the wave perturbation energy increases linearly, i.e., the shear flow is algebraically unstable due to the growth of rapid wave perturbations. This instability can be of importance in processes of turbulence development and surface and internal wave generation.     

Combined Immersed Boundary/Large-Eddy-Simulations of Incompressible Three Dimensional Complex Flows

In this paper we show how the Immersed Boundary (IB) method can be used with the Large-Eddy-Simulation (LES) to compute moderately high Reynolds number flows in complex geometric configurations. The resulting combination gives an easy-to-use, inexpensive and accurate technique which can be an important step towards the application of computational fluid dynamics (CFD) to industrially relevant problems. This paper aims at describing the main features of the method, some of the important drawbacks and possible solutions. Several representative examples are discussed in order to show the flexibility and the range of the applicability of this technique.     

Adjoint Equation-Based Methods for Control Problems in Incompressible, Viscous Flows

A review of adjoint equation-based methodologies for viscous,incompressible flow control and optimization problems is given and illustrated by a drag minimization example. A number of approaches to ameliorating the high storage and CPU costs associated with straightforward implementations of adjoint equation based methodologies are discussed. Other issues, including the relative merits of the differentiate-then-discretize and discretize-then-differentiate approaches to deriving discrete adjoint equations, the incorporation of side constraints into adjoint equation-based methodologies, and inaccuracies that occur due to differentiations at the boundary, are also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

On Plane-Parallel Separated Flows of an Incompressible Fluid near Flat Boundaries

On the basis of Stokes separated flows, examples of separated flows described by the Navier-Stokes equations of a viscous incompressible fluid are constructed. These flows are represented by series convergent in a certain non-zero neighborhood of a flat contour immersed in the flow. In this neighborhood, the series have the same structure as those for the basic Stokes flows. Examples of the regions in which the series segments chosen give only a slight deviation from the numerical solutions of the Navier-Stokes equations are presented. The comparison between inviscid separated flows (without the no-slip condition on the contour) and viscous flows of the same structure (with the no-slip condition) shows that the viscosity does not play a decisive role in the formation of separation or the type of streamline approach to or departure from the contour.     

Front Speed Enhancement by Incompressible Flows in Three or Higher Dimensions

We study, in dimensions N ≥ 3, the family of first integrals of an incompressible flow: these are ${H^{1}_{\rm loc}}$ functions whose level surfaces are tangential to the streamlines of the advective incompressible field. One main motivation for this study comes from earlier results proving that the existence of nontrivial first integrals of an incompressible flow q is the main key that leads to a “linear speed up” by a large advection of pulsating traveling fronts solving a reaction–advection–diffusion equation in a periodic heterogeneous framework. The family of first integrals is not well understood in dimensions N ≥ 3 due to the randomness of the trajectories of q and this is in contrast with the case N = 2. By looking at the domain of propagation as a union of different components produced by the advective field, we provide more information about first integrals and we give a class of incompressible flows which exhibit “ergodic components” of positive Lebesgue measure (and hence are not shear flows) and which, under certain sharp geometric conditions, speed up the KPP fronts linearly with respect to the large amplitude. In the proofs, we establish a link between incompressibility, ergodicity, first integrals and the dimension to give a sharp condition about the asymptotic behavior of the minimal KPP speed in terms of the configuration of ergodic components.     

A Priori Estimates for Free Boundary Problem of Incompressible Inviscid Magnetohydrodynamic Flows

In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions n = 2 and 3 by adopting a geometrical point of view used in Christodoulou and Lindblad (Commun Pure Appl Math 53:1536–1602, 2000), and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids.     

Large-eddy Simulation of Film Cooling Flows with Variable Density Jets

The present paper investigates the impact of the velocity and density ratio on the turbulent mixing process in gas turbine blade film cooling. A cooling fluid is injected from an inclined pipe at α=30° into a turbulent boundary layer profile at a freestream Reynolds number of Re_∞ = 400,000. This jet-in-a-crossflow (JICF) problem is investigated using large-eddy simulations (LES). The governing equations comprise the Navier–Stokes equations plus additional transport equations for several species to simulate a non-reacting gas mixture. A variation of the density ratio is simulated by the heat-mass transfer analogy, i.e., gases of different density are effused into an air crossflow at a constant temperature. An efficient large-eddy simulation method for low subsonic flows based on an implicit dual time-stepping scheme combined with low Mach number preconditioning is applied. The numerical results and experimental velocity data measured using two-component particle-image velocimetry (PIV) are in excellent agreement. The results show the dynamics of the flow field in the vicinity of the jet hole, i.e., the recirculation region and the inclination of the shear layers, to be mainly determined by the velocity ratio. However, evaluating the cooling efficiency downstream of the jet hole the mass flux ratio proves to be the dominant similarity parameter, i.e., the density ratio between the fluids and the velocity ratio have to be considered.     

Global Well-posedness of Incompressible Inhomogeneous Fluid Systems with Bounded Density or Non-Lipschitz Velocity

In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data ${a_0 \in L^\infty (\mathbb{R}^d), u_0 = (u_0^h, u_0^d) \in \dot{B}^{-1+\frac{d}{p}}_{p, r} (\mathbb{R}^d)}$ , which satisfy ${(\mu \| a_0 \|_{L^\infty} + \|u_0^h\|_{\dot{B}^{-1+\frac{d}{p}}_{p, r}}) {\rm exp}(C_r{\mu^{-2r}}\|u_0^d\|_{\dot{B}^{-1+\frac{d}{p}}_{p,r}}^{2r}) \leqq c_0\mu}$ for some positive constants c ₀, C _r and 1 < p < d, 1 < r < ∞. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity fields. Furthermore, with additional regularity assumptions on the initial velocity or on the initial density, we can also prove the uniqueness of such a solution. We should mention that the classical maximal L ^p (L ^q ) regularity theorem for the heat kernel plays an essential role in this context.     

On a Diffuse Interface Model for Two-Phase Flows of Viscous, Incompressible Fluids with Matched Densities

We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids of the same density in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. This leads to a coupled Navier–Stokes/Cahn–Hilliard system, which is capable of describing the evolution of droplet formation and collision during the flow. We prove the existence of weak solutions of the non-stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities, which ensures—in contrast to the usual case of a smooth free energy density—that the concentration stays in the physical reasonable interval. Furthermore, we find that unique “strong” solutions exist in two dimensions globally in time and in three dimensions locally in time. Moreover, we show that for any weak solution the concentration is uniformly continuous in space and time. Because of this regularity, we are able to show that any weak solution becomes regular for large times and converges as t → ∞ to a solution of the stationary system. These results are based on a regularity theory for the Cahn–Hilliard equation with convection and singular potentials in spaces of fractional time regularity as well as on maximal regularity of a Stokes system with variable viscosity and forces in L ²(0, ∞; H ^s (Ω)), ${s \in [0, \frac12)}$ , which are new themselves.