A proof of Howard's conjecture in homogeneous parallel shear flows—II: Limitations of Fjortoft's necessary instability criterion

The present paper on the linear instability of nonviscous homogeneous parallel shear flows mathematically demonstrates the correctness of Howard's [4] prediction, for a class of velocity distributions specified by a monotone functionU of the altitudey and a single point of inflexion in the domain of flow, by showing not only the existence of a critical wave numberk _c>0 but also deriving an explicit expression for it, beyond which for all wave numbers the manifesting perturbations attain stability. An exciting conclusion to which the above result leads to is that the necessary instability criterion of Fjortoft has the seeds of its own destruction in the entire range of wave numbersk>k _c—a result which is not at all evident either from the criterion itself or from its derivation and has thus remained undiscovered ever since Fjortoft enunciated [3].     

A proof of Howard’s conjecture in homogeneous parallel shear flows

A rigorous mathematical proof of Howard's conjecture which states that the growth rate of an arbitrary unstable wave must approach zero, as the wave length decreases to zero, in the linear instability of nonviscous homogeneous parallel shear flows, is presented here for the first time under the restriction of the boundedness of the second derivative of the basic velocity field with respect to the vertical coordinate in the concerned flow domain.     

On the curvature/buoyancy analogy for turbulent shear flows

Similarity in the near wall region of turbulent curved shear flows is examined. It is found that the normalized mean velocity is a function only of the dimensionless distance _c =z/L _c whereL _c is a corresponding Monin-Oboukhov length for curved shear flows. Again, the universal function is found to obey the log-linear law. Therefore, this result and the earlier derivation of So affirm that there is a very close analogy between the effects of streamline curvature and buoyancy for turbulent shear flows.

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Zusammenfassung Die Ähnlichkeitsverhältnisse in turbulenten, gekrümmten Strömungen mit Schubkräften wird für das Gebiet in der Nähe einer Wand untersucht. Es ergibt sich, daß die normalisierte mittlere Geschwindigkeit nur von der dimensionslosen Entfernung _c =z/L _c abhängt.L _c ist hierbei eine zugeordnete Monin-Oboukhov-Länge für gekrümmte Strömungen mit Schubkräften. Auch in diesem Falle geohorcht die allgemeine Gleichung dem logarithmisch-linearen Gesetz. Dieses Ergebnis und die frühere Ableitung von So bestätigen, daß eine ausgeprägte Analogie zwischen den Auswirkungen der Strömungslinienkrümmung und dem Auftrieb bei turbulenten Strömungen mit Schubkräften besteht.

     

On heat transfer modelling for turbulent shear flows on curved surfaces

Summary A second-order closure method for turbulent flow has been used by Dr. R. M. C. So to predict the effects of wall curvature on boundary layer heat transfer, apparently with good results. In the present paper it is shown that these good results are attributable mainly to algebraic errors made in working out the analysis and partly to unrealistic values having been assigned to the model coefficients. Attention is drawn to some of the fundamental difficulties encountered in modelling turbulent heat transfer and it is concluded that these have yet to be overcome for complex flows.

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Zusammenfassung Eine Schließungsmethode zweiter Ordnung für turbulente Strömungen wurde von Dr. R. M. C. So benützt um Effekte der Wandkrümmung auf die Wärmeübertragung vorauszusagen, mit anscheinend guten Resultaten. In dieser Arbeit wird gezeigt, daß diese guten Ergebnisse hauptsächlich gewissen algebraischen Fehlern in der Analyse zuzuschreiben sind; zum Teil sind auch den Modell-Koeffizienten unrealistische Werte zugeordnet worden. Es wird auf die grundsätzlichen Schwierigkeiten hingewiesen, die bei Modellen für turbulente Wärmeübertragung auftreten, und es wird der Schluß gezogen, daß diese für komplizierte Strömungen noch nicht überwunden worden sind.

     

Logarithm laws for flows on homogeneous spaces

In this paper we generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a sequence of subsets {A _t  | t∈ℕ} of a homogeneous space G/Γ (G a semisimple Lie group, Γ an irreducible lattice) and a sequence of elements f _t of G under which #{t∈ℕ | f _t x∈A _t } is infinite for a.e. x∈G/Γ. The main tool is exponential decay of correlation coefficients of smooth functions on G/Γ. Besides the general (higher rank) version of Sullivan’s result, as a consequence we obtain a new proof of the classical Khinchin-Groshev theorem on simultaneous Diophantine approximation, and settle a conjecture recently made by M. Skriganov. Oblatum 27-VII-1998 & 2-IV-1999 / Published online: 5 August 1999     

On the spectrum of shear flows and uniform ergodic theorems

The spectra of parallel flows (that is, flows governed by first-order differential operators parallel to one direction) are investigated, on both L²$L^2$ spaces and weighted-L²$L^2$ spaces. As a consequence, an example of a flow admitting a purely singular continuous spectrum is provided. For flows admitting more regular spectra the density of states is analyzed, and spaces on which it is uniformly bounded are identified. As an application, an ergodic theorem with uniform convergence is proved.     

Stability of stratified shear flows

For a steady plane parallel flow of an inviscid, incompressible fluid of variable density under gravity, it is shown that the complex wave velocity for any unstable mode lies in a semiellipse-type region whose major axis coincides with the diameter of Howard's semicircle, while its minor axis depends on the stratification. If kc_i denotes the complex part of wave frequency and J₀ the minimum of the local Richardson number over the flow domain, it is further established that kc_i → 0+ as $\text{J}{}_0\text{→}\text{1}\text{4}\text{?}$. The case of free upper surface and conditional reduction dependent on the curvature of the basic velocity of the unstable region is also studied.     

On the stability of shear flows of Prandtl type for the steady Navier-Stokes equations

Science China Mathematics - In this paper, we study the stability of shear flows of Prandtl type as $$left({Uleft({y/sqrt nu} right),0} right)$$ for the steady Navier-Stokes equations under a...     

Barotropic instability of shear flows

We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm-Liouville theory and hypergeometric functions.     

On the global regularity of shear thinning flows in smooth domains

In some recent papers we have been pursuing regularity results up to the boundary, in W2,l(Ω) spaces for the velocity, and in W1,l(Ω) spaces for the pressure, for fluid flows with shear dependent viscosity. To fix ideas, we assume the classical non-slip boundary condition. From the mathematical point of view it is appropriate to distinguish between the shear thickening case, p>2, and the shear thinning case, p<2, and between flat-boundaries and smooth, arbitrary, boundaries. The p<2 non-flat boundary case is still open. The aim of this work is to extend to smooth boundaries the results proved in reference [H. Beirão da Veiga, On non-Newtonian p-fluids. The pseudo-plastic case, J. Math. Anal. Appl. 344 (1) (2008) 175-185]. This is done here by appealing to a quite general method, introduced in reference [H. Beirão da Veiga, On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem, J. Eur. Math. Soc., in press], suitable for considering non-flat boundaries.     

Poisson structures for geometric curve flows in semi-simple homogeneous spaces

We apply the equivariant method of moving frames to investigate the existence of Poisson structures for geometric curve flows in semi-simple homogeneous spaces. We derive explicit compatibility conditions that ensure that a geometric flow induces a Hamiltonian evolution of the associated differential invariants. Our results are illustrated by several examples of geometric interest.     

Partial regularity for weak heat flows into riemannian homogeneous spaces

It is proved that any weak flow of harmonic maps into a compact homogeneous manifold satisfying the monotonicity inequality and the energy inequality is regular off a closed set of m-dimensional Hausdorff measure zero (w.r.t parabolic metric), and coincides with a regular flow if the latter one exists. Moreover, it is also shown that the weak limit of a sequence of such weak flows is a weak flow.     

Unique ergodicity of flows on homogeneous spaces

LetG be a unimodular Lie group, Γ a co-compact discrete subgroup ofG and ‘a’ a semisimple element ofG. LetT _a be the mapgΓ →ag Γ:G/Γ →G/Γ. The following statements are pairwise equivalent: (1) (T _a, G/Γ,θ) is weak-mixing. (2) (T _a, G/Γ) is topologically weak-mixing. (3) (G ^u, G/Γ) is uniquely ergodic. (4) (G ^u, G/Γ,θ) is ergodic. (5) (G ^u, G/Γ) is point transitive. (6) (G ^u, G/Γ) is minimal. If in additionG is semisimple with finite center and no compact factors, then the statement “(T _a, G/Γ,θ) is ergodic” may be added to the above list. The authors were partially supported by NSF grant MCS 75-05250.     

Elliptical flows perturbed by shear waves

We consider the superimposition of two shear waves on a pseudo-plane motion of the first kind with elliptical streamlines. If the shear waves satisfy some special assumptions it is possible to establish a recurrence relation among the Rivlin–Ericksen tensors associated with the flow at hand. This remarkable kinematical result allows to determine new exact solutions for a large class of materials and to generalize some well known solutions modelling special flows (such as the celebrated Berker’s solution for a Navier–Stokes fluid in an orthogonal rheometer).     

Heat transfer modeling for turbulent shear flows on curved surfaces

This paper examines two models for two-dimensional curved flow heat transfer. Both models are based on the assumption of local-equilibrium turbulence. Yet, one model predicts an increase for the turbulent Prandtl number with convex curvature, while the other gives the opposite behavior. Through examination of the near wall temperature profiles, it is possible to identify the correct behavior for turbulent Prandtl number. It is found that the turbulent Prandtl number for curved flows increases with convex curvature. The discrepancy between the two models is traced to the modeling terms proposed for the pressure-temperature-gradient correlation and for wall corrections. While the introduction of the rapid component and the wall corrections in the modeling of the pressure-strain correlation do not affect the behavior of the resultant shear stress, these modeling terms in the pressure-temperature-gradient correlation cause the turbulent Prandtl number to decrease with convex curvature.

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Zusammenfassung In dieser Arbeit werden zwei Modelle für zweidimensionale Wärmeübertragung in gekrümmten Strömungen untersucht. Beide Modelle basieren auf der Annahme von lokalem Gleichgewicht innerhalb der turbulenten Strömung. Trotzdem resultiert das eine Modell in einer Erhöhung der turbulenten Prandtl-Zahl für konvexe Krümmungen, während das zweite Modell genau das Gegenteil vorhersagt. Es ist möglich, durch Untersuchung des Temperaturprofiles in Wandnähe das korrekte Verhalten der turbulenten Prandtl-Zahl zu identifizieren. Es stellt sich dabei heraus, daß die turbulente Prandtl-Zahl für gekrümmte Strömungen in konvexen Kurven zunimmt. Der Widerspruch zwischen den beiden Modellen läßt sich auf die Abbildungsausdrücke zurückführen, die für die Wechselbeziehungen zwischen den Druck- und Temperaturgradienten und die für die Wandkorrekturen angesetzt worden sind. Während die Einführung der schnellen Komponente und die Wandkorrekturen in der Darstellung der Wechselbeziehung zwischen Druck und Beanspruchung keinen Einfluß auf das Verhalten des resultierenden Schubdruckes haben, verursachen diese Darstellungsausdrücke für die Wechselbeziehung zwischen Druck- und Temperaturgradienten ein Abnehmen der turbulenten Prandtl-Zahl für konvexe Kurven.

     

On the instability of internal Alfvén-gravity waves in stratified shear flows

Internal Alfvén-gravity waves and their stability characteristics for an inviscid, nondissipative, Boussinesq fluid undergoing shear in the presence of a lower rigid boundary is studied. The model consists of a layer of constant shear capped by a layer of constant velocity. The Brunt-Väisälä frequency is assumed to be uniform throughout the fluid. The unstable modes have wavelenghts which are close to those of the internal Alfvén-gravity waves which propagate from the troposphere into the ionosphere.