Prandtl's Layer Research Articles

Validity of steady Prandtl layer expansions

AbstractLet the viscosity for the 2D steady Navier‐Stokes equations in the region and with no slip boundary conditions at . For , we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. Our uniform estimates in ε are achieved through a fixed‐point scheme: for solving the Navier‐Stokes equations, where are the tangential and normal velocities at , DNS stands for of the vorticity equation for the normal velocity v, and the compatibility ODE for at .

On the dynamics of the near-surface currents in the Arctic Ocean

Starting from the Navier–Stokes equation (in rotating, spherical coordinates), and the equation of state coupled with the first law of thermodynamics for the atmosphere, a general description of the near-surface flow in the Arctic Ocean, and the atmosphere above it, is developed. This includes a suitable stress boundary condition at the air–ice–water interface. Invoking only the thin-shell approximation, for both the atmosphere and the ocean, an appropriate asymptotic reduction is developed; this approach ensures that all the physical attributes are retained at leading order. The wind is described via the properties of the Prandtl layer and the atmospheric boundary layer, with some observations about the geostrophic flow higher in the troposphere. The corresponding equations for the ocean, with a suitable surface wind, are used to model the two dominant features of the arctic surface-current field: the rotating Beaufort Gyre and, using a transformation of the spherical coordinates that removes the polar singularities in geographical coordinates, the Transpolar Drift. In addition, we show that the stress conditions at the ice–water interface ensure that the sea-ice necessarily moves faster than the ocean on which it floats, and in a direction between that of the surface wind and that of the ocean surface current.

Steady Prandtl layer expansions with external forcing

In this article we apply the machinery developed by Guo and Iyer [Validity of Steady Prandtl Layer Expansions, Comm. Pure Appl. Math. (to appear) (2022)] together with a new compactness estimate and a new quantity, the “degree”, in order to prove validity of steady Prandtl layer expansions with external forcing. The compactness techniques introduced in this article allow us to treat more general background Prandtl layers than Guo and Iyer [Validity of Steady Prandtl Layer Expansions, Comm. Pure Appl. Math. (to appear) (2022)].

Regularity and Expansion for Steady Prandtl Equations

Due to degeneracy near the boundary, the question of high regularity for solutions to the steady Prandtl equations has been a longstanding open question since the celebrated work of Oleinik. We settle this open question in affirmative in the absence of an external pressure. Our method is based on energy estimates for the quotient, $$q = \frac{v}{\bar{u}}$$ , $$\bar{u}$$ being the classical Prandtl solution, via the linear derivative Prandtl (LDP) equation. As a consequence, our regularity result leads to the construction of Prandtl layer expansion up to any order.

Validity of Prandtl layer theory for steady magnetohydrodynamics over a moving plate with nonshear outer ideal MHD flows

In this paper, we validate the boundary layer theory for 2D steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain {(X,Y)∈[0,L]×R+} under the assumption of a moving boundary at {Y=0}. The validity of boundary layer expansions and convergence rates are established in Sobolev sense. We extend the results for the case with the shear outer ideal MHD flows [3] to the case of the nonshear flows.

On Global-in-x Stability of Blasius Profiles

We characterize the well known self-similar Blasius profiles, $$[{\bar{u}}, {\bar{v}}]$$, as downstream attractors to solutions [u, v] to the 2D, stationary Prandtl system. It was established in Serrin (Proc R Soc Lond A 299:491–507, 1967) using maximum principle techniques that $$\Vert u - {\bar{u}}\Vert _{L^\infty _y} \rightarrow 0$$ as $$x \rightarrow \infty $$. In the case of localized data near Blasius, this paper provides an energy based proof of asymptotic stability. Central to our analysis is a new weighted “quotient estimate” which couples with a higher order, nonlinear energy cascade. Similar quotient estimates have played a crucial role in establishing the validity of the inviscid Prandtl layer expansion in Guo and Iyer (Validity of steady Prandtl layer expansions. arXiv:1805.05891 2018).

Steady Prandtl Layers Over a Moving Boundary: Nonshear Euler Flows

In this article we establish the validity of Prandtl layer expansions around Euler flows which are not shear. The presence of nonshear flows at the leading order creates a scaling of $\mathcal{O}(\...

Analysis of Carreau fluid in the presence of thermal stratification and magnetic field effect

Abstract In current article theoretical analysis is executed to examine flow features of Carreau fluid by conferring scintillating aspects of thermal stratification. Flow field equations are attained by incorporating infinite shear rate viscosity and magnetic field effects. Afterwards, the partial differential obtained from the fundamental laws of continuity, momentum and energy containing stratification aspects are attained. These consequent partial differential expressions are so intricate that it seems difficult to solve analytically. Therefore Prandtl layer approximation is capitalized to retain efficient parts of flow narrating differential equations. Then next step is to eradicate the in active parts of partial equations by implementing transformations. The solution structured of reduced system is acquired by self-coded shooting algorithm. The variation in physical profiles with respect to involved parameters is exhibited with the aid of graphs and tables. It is inferred that Carreau fluid behaves in opposite pattern for n > 1 (shear thickening) and n

Sublayer of Prandtl Boundary Layers

The aim of this paper is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit: $\nu \to 0$. In \cite{Grenier}, one of the authors proved that there exists no asymptotic expansion involving one Prandtl's boundary layer with thickness of order $\sqrt\nu$, which describes the inviscid limit of Navier-Stokes equations. The instability gives rise to a viscous boundary sublayer whose thickness is of order $\nu^{3/4}$. In this paper, we point out how the stability of the classical Prandtl's layer is linked to the stability of this sublayer. In particular, we prove that the two layers cannot both be nonlinearly stable in $L^\infty$. That is, either the Prandtl's layer or the boundary sublayer is nonlinearly unstable in the sup norm.

Featuresof velocity distribution in a turbulent flow

In this article the questions of kinematic structure of steady turbulent flow near a solid boundary are considered. It has been established that due to friction the value of the local Reynolds number decreases and always becomes smaller than the critical value of the Reynolds number, which leads to formation of viscous flow near a wall. Velocity profiles for the area of viscous flow with constant and variable shear stress are obtained. The experimental investigations of different authors showed that in this area the flow is of unsteady character, where viscous flow occurs intermittently with turbulent flow. With increasing distance from the wall the flow becomes fully turbulent. In the area where generation and dissipation of turbulence are very intensive, there is a developed turbulent flow with increasing distance from the wall. Dissipation of turbulence is an action of viscous force. The logarithmic velocity profile was obtained by L. Prandtl disregarding the viscous component and the linear variation of the shear stress in the depth flow. The profile parameters C and k were determined from Nikuradze’s experiments. The detailed investigations of Nikuradze’s experiments established the part of the flow where the logarithmic velocity profile is correctly confirmed.This part of the flow was called “Prandtl layer”. The measured velocity distribution above this layer deviates in the direction of greater values. Processing of experimental data revealed that the thickness of the “Prandtl layer”, normalized to the radius of a pipe, depend on a drag coefficient. The formula for determining the thickness of the “Prandtl layer” with the known value of the drag coefficient is obtained. It is shown that the thickness of “Prandtl layer” almost coincides with the boundary layer displacement thickness formed on the wall of the pipe.

Boundary-layer interactions in the plane-parallel incompressible flows

We study the inviscid limit problem of incompressible flows in the presence of both impermeable regular boundaries and a hypersurface transversal to the boundary across which the inviscid flow has a discontinuity jump. In the former case, boundary layers have been introduced by Prandtl as correctors near the boundary between the inviscid and viscous flows. In the latter case, the viscosity smoothes out the discontinuity jump by creating a transition layer which has the same amplitude and thickness as the Prandtl layer. In the neighbourhood of the intersection of the impermeable boundary and of the hypersurface, interactions between the boundary and the transition layers must then be considered. In this paper, we initiate a mathematical study of this interaction and carry out a strong convergence in the inviscid limit for the case of the plane-parallel flows introduced by Di Perna and Majda (1987 Commun. Math. Phys. 108 667–89).

Rotating stratified flow past a steep-sided obstacle: incipient separation

Many of the most interesting phenomena observed to occur in the flow of rotating and stratified fluids past obstacles, for example eddy shedding and wake unsteadiness, are due to separation of the boundary layer on the obstacle or its Taylor column. If the Rossby number of the flow lies between E½ and E (E is the Ekman number) and the Burger number is small, the structure of a viscous shear layer of width E⅙ on the circumscribing cylinder of an axisymmetric obstacle controls the inviscid flow. The surface boundary layer is not an Ekman layer, but a Prandtl layer, even at small Rossby numbers. As the slope of the obstacle at its base increases, the nature of the inviscid motion is altered substantially, in the rotation-dominated regime. We show that, for sufficiently large slopes, the flow develops a small region of non-uniqueness external to the column, simultaneously with the separation of the narrow band of fluid flowing round the base of the object.

Coupled oceanic and atmospheric Ekman layers in the tropics adapted to the central pacific ocean

An idealizing analytic solution of the latitudinally differentiated equation of horizontal motion is applied. It neglects temporal and zonal variations and it supposes the zonally averaged distributions of pressure to be prescribed. The solution couples both the Ekman layers by the surface stress being assumed proportionally to the square of the difference of velocities of sea and air at the Prandtl layer. Thermal effects of the air-sea interaction are ignored. The resulting winds and currents of a numerical example adapted to the Pacific Ocean allow for a continuous passage to mid-latitudes and between the hemispheres, and they show essential properties of the scheme of surface currents. Moreover a “South Equatorial Counter Current” in a southern zone of doldrums can be induced without the company of a westward ascent of the sea surface but in the case of lacking westward wind stress and physically substantiated latitudinal variation of the sea surface level. The model yields the occuring “Southern Doldrums” (or even weak westerly winds) in that near equatorial zone where the northward gradient of surface air pressure reduces to its value at the equator (or falls short of it, respectively).

A three dimensional Prandtl layer conception for air currents over moving rough water surfaces

The usual determinations of wind stress and surface drag coefficient from wind speed observations in the maritime Prandtl layer ignore the traction effects of surface currents, of sea and roughened swell — and, consequently, a possible veering of the mean wind with height. A conception, which allows for these effects and converges to the known one if the mean wind at the sea surface vanishes (or at least has the direction of the wind in the height of 10 m) is deduced on nearly the same basis as Prandtl's theory. With the effects taken into consideration, a part of the large scatter of the drag coefficients evaluated is accounted for.

Turbulent hydrodynamic voltammetry : Part II. Investigations of turbulence frequency in a hydrodynamic voltammetric cell

A new relationship has been developed for the interpretation of turbulence data for voltammetric signals measured in turbulently flowing liquids. The electric current corresponding to the ripples at the Prandtl layer can be described by the equation ĩ df ( v) = K v −1, where v is the frequency of the fluctuating electric current, which is proportional to the frequency of ripple formation. The kinetic energy of the ripples contained in unit mass is constant for all frequencies of the ripples.

Heat transfer in a supersonic flow

At present it is commonly assumed that some peculiar surface exists which bounds a viscous region around a body in a supersonic flow. This region is identified with a laminar or turbulent Prandtl layer, the process of transition of visible motion into heat being described by the procedures developed by Prandtl and von Kármán for subsonic flows. In the present work another approach is used based on the ideas of Osborne Reynolds and the so-called resolution equation where transition of thermal motion into pulsating one is fixed which, in its turn establishes relationship between the Nusselt and Reynolds numbers. This is a power-law relationship in which the coefficients and power exponent may be pre-calculated based on simple considerations. This has been done in the present work.

Martian winds and dust clouds

Previous calculations of the surface wind stress required to raise dust on Mars are reconsidered and the threshold friction velocity is found to be about 2.0 m sec −1 with particles of 200–300 μm being the most easily lifted. With this friction velocity, the planetary resistance law yields a corresponding wind at the top of the Ekman layer of 60 m sec −1, and the logarithmic wind law yields a corresponding wind at the top of the Prandtl layer of 38 m sec −1. These speeds are somewhat lower than those used by previous investigators. Various mechanisms for producing such strong winds are examined and it is concluded that the general circulation, thermal effects of topography, mechanical effects of topography and dust devils are all capable of doing so. Dust storms associated with small-scale disturbances are found to be incapable of growth. A scaling analysis of the equations of horizontal motion and of hydrostatic balance shows that a dust cloud at least 10 km thick and several tens of km in radius can, by absorption of sunlight, generate temperature gradients that, in turn, produce winds capable of raising more dust. Thus, a feedback mechanism is suggested in which an initial dust cloud exceeding certain critical dimensions can grown to planetary size. The preference of large dust storms to occur at southern hemisphere summer solstice is attributed to the maximum of insolation at that time. It is suggested that the frequent origin in the Noachis-Hellas region may be due to orographic features of the right scale and to low height in that area.

A NUMERICAL EXPERIMENT ON THE DEVELOPMENT OF A TROPICAL CYCLONE

Abstract A model of a tropical cyclone is constructed which is based upon conservation of momentum, mass,water vapor and heat in the hydrostatic system. The horizontal and vertical eddy-exchange processes for momentum, moisture and heat are included in the equations in order to incorporate the planetary frictional (Ekman) layer into the model. The effects of the surface boundary (Prandtl) layer are simulated by the boundary conditions for the equations, which permit the evaluation of surface stress, the sensible heat transport and the evaporation of water vapor from the earth surface. The energy sources of the model are the latent heat of condensation released during the ascent of moist air and the sensible heat transported from the ocean surface. The formulation of the finite-difference equations for the axially-symmetric case is presented, together with an examination of the computational stability. By means of a high-speed computer, two independent computations with and without the supply of latent hea...