Plate-injection into a separated supersonic boundary layer. Part 2. The transition regions

1974 ◽  
Vol 62 (2) ◽  
pp. 289-304 ◽  
Author(s):  
K. Stewartson
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Flow Field ◽  
Asymptotic Expansions ◽  
Stokes Equations ◽  
Supersonic Boundary Layer ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Consistent Solution ◽  
Strong Injection

The model proposed by Smith & Stewartson (1973), to describe the separated boundary layer induced by strong injection over a finite length of a flat plate in a supersonic mainstream, is shown to provide the basis for a fully consistent solution of the Navier–Stokes equations for this problem, valid in the limit of infinite Reynolds number. The solution takes the form of asymptotic expansions in each of a large number of overlapping regions of the flow field, which are consistently matched across areas of common validity.

Free-stream coherent structures in parallel boundary-layer flows

2014 ◽  
Vol 752 ◽  
pp. 602-625 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Coherent Structures ◽  
Free Stream ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Homotopy Continuation ◽  
Boundary Layer Flows ◽  
Navier Stokes Equations ◽  
High Reynolds

AbstractOur concern in this paper is with high-Reynolds-number nonlinear equilibrium solutions of the Navier–Stokes equations for boundary-layer flows. Here we consider the asymptotic suction boundary layer (ASBL) which we take as a prototype parallel boundary layer. Solutions of the equations of motion are obtained using a homotopy continuation from two known types of solutions for plane Couette flow. At high Reynolds numbers, it is shown that the first type of solution takes the form of a vortex–wave interaction (VWI) state, see Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666), and is located in the main part of the boundary layer. On the other hand, here the second type is found to support an equilibrium solution of the unit-Reynolds-number Navier–Stokes equations in a layer located a distance of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}O(\ln \mathit{Re})$ from the wall. Here $\mathit{Re}$ is the Reynolds number based on the free-stream speed and the unperturbed boundary-layer thickness. The streaky field produced by the interaction grows exponentially below the layer and takes its maximum size within the unperturbed boundary layer. The results suggest the possibility of two distinct types of streaky coherent structures existing, possibly simultaneously, in disturbed boundary layers.

Characterization of coherent vortical structures in a supersonic turbulent boundary layer

2008 ◽  
Vol 613 ◽  
pp. 205-231 ◽  
Author(s):  
SERGIO PIROZZOLI ◽  
MATTEO BERNARDINI ◽  
FRANCESCO GRASSO
Keyword(s):  
Boundary Layer ◽  
Outer Layer ◽  
Stokes Equations ◽  
Supersonic Boundary Layer ◽  
Navier Stokes ◽  
Hairpin Vortices ◽  
Length Scales ◽  
Navier Stokes Equations ◽  
Vortical Structures

A spatially developing supersonic boundary layer at Mach 2 is analysed by means of direct numerical simulation of the compressible Navier--Stokes equations, with the objective of quantitatively characterizing the coherent vortical structures. The study shows structural similarities with the incompressible case. In particular, the inner layer is mainly populated by quasi-streamwise vortices, while in the outer layer we observe a large variety of structures, including hairpin vortices and hairpin packets. The characteristic properties of the educed structures are found to be nearly uniform throughout the outer layer, and to be weakly affected by the local vortex orientation. In the outer layer, typical core radii vary in the range of 5–6 dissipative length scales, and the associated circulation is approximately constant, and of the order of 180 wall units. The statistical properties of the vortical structures in the outer layer are similar to those of an ensemble of non-interacting closed-loop vortices with a nearly planar head inclined at an angle of approximately 20° with respect to the wall, and with an overall size of approximately 30 dissipative length scales.

Impulsively started flow separation in wavy-walled tubes

1998 ◽  
Vol 359 ◽  
pp. 1-22 ◽  
Author(s):  
FEDERICO DOMENICHINI ◽  
GIANNI PEDRIZZETTI
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Stokes Equations ◽  
Reverse Flow ◽  
Adverse Pressure Gradient ◽  
Boundary Layer Separation ◽  
Navier Stokes ◽  
Asymptotic Model ◽  
Navier Stokes Equations ◽  
Boundary Layer Problem

The axisymmetric boundary-layer separation of an incompressible impulsively started flow in a wavy-walled tube is analysed at moderate to high values of the Reynolds number. The investigation is carried out by numerical integration of either the Navier–Stokes equations or Prandtl's asymptotic formulation of the boundary-layer problem. The presence of an adverse pressure gradient induces reverse flow at the tube wall independently of the Reynolds number; its occurrence can be predicted by a timescale analysis. Following that, the viscous calculations show different dynamics depending on the Reynolds number. As the Reynolds number increases, the boundary layer has in a well-defined internal structure where longitudinal lengthscales become comparable with the viscous one. Thus the boundary-layer scaling fails locally, with a minimum of pressure inside the boundary layer itself. The formation of the primary recirculation is well captured by the asymptotic model which, however, is not able to describe the roll-up of the vortex structure inside the recirculating region. This inadequacy appears well before the flow evolves to the characteristic terminal singularity usually assumed as foreshadowing the vortex shedding phenomenon. The outcomes are compared with the existing results of analogous problems giving an overall agreement but improving, in some cases, the physical picture.

scholarly journals Derivation of a Viscous Serre–Green–Naghdi Equation: An Impasse?

Fluids ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 135
Author(s):  
Denys Dutykh ◽  
Hervé V.J. Le Meur
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Euler Equations ◽  
Stokes Equations ◽  
Current Status ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Bottom Region ◽  
Flow Domain ◽  
Layer Flow

In this article, we present the current status of the derivation of a viscous Serre–Green–Naghdi system. For this goal, the flow domain is separated into two regions. The upper region is governed by inviscid Euler equations, while the bottom region (the so-called boundary layer) is described by Navier–Stokes equations. We consider a particular regime binding the Reynolds number and the shallowness parameter. The computations presented in this article are performed in the fully nonlinear regime. The boundary layer flow reduces to a Prandtl-like equation that we claim to be irreducible. Further approximations are necessary to obtain a tractable model.

scholarly journals A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equations

2021 ◽  
Author(s):  
Taofiq Amoloye
Keyword(s):  
Reynolds Number ◽  
Numerical Simulations ◽  
Potential Theory ◽  
Stream Function ◽  
Continuity Equation ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Consistent Solution

Abstract The three main approaches in fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics.Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author refined CPT. He introduced refined potential theory (RPT) that provides a viscous potential/stream function as a physically consistent solution to the NSE problem. This function captures observable unsteady flow features including separation, wake, vortex shedding, compressibility, turbulence, and Reynolds-number-dependence. It appropriately combines the properties of a three-dimensional potential function that satisfy the inertia terms of NSE and the features of a stream function that satisfy the continuity equation, the viscous vorticity equation, and the viscous terms of NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere, and spheroid.

scholarly journals Spectral stability of Prandtl boundary layers: An overview

Analysis ◽  
2015 ◽  
Vol 35 (4) ◽  
Author(s):  
Emmanuel Grenier ◽  
Yan Guo ◽  
Toan T. Nguyen
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Asymptotic Expansions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Stability Of Shear Flows ◽  
The Stability ◽  
Prandtl Boundary Layer ◽  
Physical Instability

AbstractIn this paper we show how the stability of Prandtl boundary layers is linked to the stability of shear flows in the incompressible Navier–Stokes equations. We then recall classical physical instability results, and give a short educational presentation of the construction of unstable modes for Orr–Sommerfeld equations. We end the paper with a conjecture concerning the validity of Prandtl boundary layer asymptotic expansions.

scholarly journals Numerical simulation on the development of periodic three-dimensional disturbances in a supersonic boundary layer

2019 ◽  
Vol 196 ◽  
pp. 00016
Author(s):  
Gleb Kolosov ◽  
Alexander Semenov ◽  
Alexey Yatskikh
Keyword(s):  
Boundary Layer ◽  
Mass Flow ◽  
Stokes Equations ◽  
Numerical Study ◽  
Three Dimensional ◽  
Small Diameter ◽  
Supersonic Boundary Layer ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Software Complex

The results of a numerical study of the development of periodic pulsations in a supersonic boundary layer on a flat plate are presented at a Mach number of 2.5 and a unit Reynolds number of 8×106 m–1. Using the software complex ANSYS, the complete Navier-Stokes equations were solved. Periodic mass flow disturbances with a frequency of 20 kHz were introduced into the boundary layer through a small-diameter hole on the surface of the model. Downstream the profiles of the longitudinal mass flow pulsations were recorded, and spectral analysis of the data was carried out. The main characteristics of the development of unstable disturbances in both physical and wave spaces are obtained.

scholarly journals Numerical Simulation of Disturbance Evolution in the Supersonic Boundary Layer over an Expansion Corner

2021 ◽  
Vol 56 (5) ◽  
pp. 645-656
Author(s):  
P. V. Chuvakhov ◽  
I. V. Egorov
Keyword(s):  
Numerical Simulation ◽  
Boundary Layer ◽  
Stokes Equations ◽  
Supersonic Boundary Layer ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Flow Stabilization ◽  
Stabilization Effect ◽  
Aerodynamic Experiment ◽  
Linear And Nonlinear

Abstract— The linear and nonlinear stages of disturbance development in the supersonic boundary layer over a 10° expansion corner is investigated numerically within the framework of Navier—Stokes equations for Mach number 3. The effect of sudden flow expansion on the disturbance evolution is analyzed. The flow stabilization effect observable in the aerodynamic experiment is also discussed.

Numerical Integration of the Navier-Stokes Equations for a Disk Rotating in a Housing

1985 ◽  
Vol 40 (8) ◽  
pp. 789-799 ◽  
Author(s):  
A. F. Borghesani
Keyword(s):  
Boundary Layer ◽  
Cylindrical Cavity ◽  
Boundary Layer Thickness ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Rotating Disk ◽  
Infinite Medium ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Viscous Torque

The Navier-Stokes equations for the fluid motion induced by a disk rotating inside a cylindrical cavity have been integrated for several values of the boundary layer thickness d. The equivalence of such a device to a rotating disk immersed in an infinite medium has been shown in the limit as d → 0. From that solution and taking into account edge effect corrections an equation for the viscous torque acting on the disk has been derived, which depends only on d. Moreover, these results justify the use of a rotating disk to perform accurate viscosity measurements.

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

Sign in / Sign up

Export Citation Format

Share Document