Boundary Layers With Viscous Potential Outer Flows

Volume 1 ◽  
2004 ◽  
Author(s):  
Alireza Najafiyazdi
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Additional Term ◽  
Reynolds Numbers ◽  
Momentum Equation ◽  
Outer Flow ◽  
Bernoulli’S Equation ◽  
Momentum Integral

In this paper we try to derive improved formulations for laminar boundary layers in incompressible flows by using the concept of viscous potential flows, presented in Joseph [1] and Joseph [2], as the outer flow. Bernoulli’s equation is used at the edge of boundary layer to make the basic assumption to deduce the presented formulas. The momentum equation is derived directly from the Navier-Stokes equations and then simplified using an order of magnitude analyze. However an additional term, μ∂2u∞∂x2, remains in the momentum equation to represent the contribution of the viscosity of the outer potential flow at the edge of the boundary layer. The viscosity of the outer viscous flow shows itself also in momentum-integral and energy-integral equations. Numerical results showed that at high Reynolds numbers and low angles of attack the results of the two formulas are almost the same, but for lower Reynolds numbers and higher angles of attack the difference between the results is remarkable.

Essential Ingredients for the Computation of Steady and Unsteady Blade Boundary Layers

1991 ◽  
Vol 113 (4) ◽  
pp. 608-616 ◽  
Author(s):  
H. M. Jang ◽  
J. A. Ekaterinaris ◽  
M. F. Platzer ◽  
T. Cebeci
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Transitional Region ◽  
Low Reynolds Numbers ◽  
Flow Equations ◽  
Low Reynolds

Two methods are described for calculating pressure distributions and boundary layers on blades subjected to low Reynolds numbers and ramp-type motion. The first is based on an interactive scheme in which the inviscid flow is computed by a panel method and the boundary layer flow by an inverse method that makes use of the Hilbert integral to couple the solutions of the inviscid and viscous flow equations. The second method is based on the solution of the compressible Navier–Stokes equations with an embedded grid technique that permits accurate calculation of boundary layer flows. Studies for the Eppler-387 and NACA-0012 airfoils indicate that both methods can be used to calculate the behavior of unsteady blade boundary layers at low Reynolds numbers provided that the location of transition is computed with the en method and the transitional region is modeled properly.

scholarly journals Essential Ingredients for the Computation of Steady and Unsteady Blade Boundary Layers

1990 ◽  
Author(s):  
H. M. Jang ◽  
M. F. Platzer ◽  
J. A. Ekaterinaris ◽  
T. Cebeci
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Transitional Region ◽  
Low Reynolds Numbers ◽  
Flow Equations ◽  
Low Reynolds

Two methods are described for calculating pressure distributions and boundary layers on blades subjected to low Reynolds numbers and ramp–type motion. The first is based on an interactive scheme in which the inviscid flow is computed by a panel method and the boundary layer flow by an inverse method that makes use of the Hilbert integral to couple the solutions of the inviscid and viscous flow equations. The second method is based on the solution of the compressible Navier–Stokes equations with an embedded grid technique that permits accurate calculation of boundary layer flows. Studies for the Eppler and NACA–0012 airfoils indicate that both methods can be used to calculate the behavior of unsteady blade boundary layers at low Reynolds numbers provided that the location of transition is computed with the en–method and the transitional region is modelled properly.

scholarly journals Experiments on the structure and scaling of hypersonic turbulent boundary layers

2017 ◽  
Vol 834 ◽  
pp. 237-270 ◽  
Author(s):  
Owen J. H. Williams ◽  
Dipankar Sahoo ◽  
Mark L. Baumgartner ◽  
Alexander J. Smits
Keyword(s):  
Boundary Layer ◽  
Mach Number ◽  
Boundary Layers ◽  
Rayleigh Scattering ◽  
Incompressible Flows ◽  
Reynolds Numbers ◽  
Correlation Lengths ◽  
Velocity Statistics ◽  
Incompressible Boundary ◽  
Free Stream Mach Number

Particle image velocimetry and filtered Rayleigh scattering experiments were performed over a range of Reynolds numbers to study the scaling and structure of a smooth, flat-plate turbulent boundary layer with a free stream Mach number of 7.5. The measurements indicate few, if any, dynamic differences due to Mach number. Mean and fluctuating streamwise velocities in the outer layer show strong similarity to incompressible flows at comparable Reynolds numbers when scaled according to van Driest and Morkovin. In addition, correlation lengths and structure angles based on velocity statistics were found to be less sensitive to compressibility than indicated by previous studies based on density fields or mass-weighted statistics, suggesting that the density and velocity fields obey different scaling. Finally, the boundary layer displays uniform momentum zones, with the number of these zones similar to incompressible boundary layers at comparable Reynolds numbers.

The Departure from Equilibrium of Turbulent Boundary Layers

1968 ◽  
Vol 19 (1) ◽  
pp. 1-19 ◽  
Author(s):  
H. McDonald
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Kinetic Energy ◽  
Stress Distribution ◽  
Boundary Layers ◽  
Turbulent Boundary Layers ◽  
Two Dimensional ◽  
Pressure Distributions ◽  
Momentum Integral ◽  
State Examination

SummaryRecently two authors, Nash and Goldberg, have suggested, intuitively, that the rate at which the shear stress distribution in an incompressible, two-dimensional, turbulent boundary layer would return to its equilibrium value is directly proportional to the extent of the departure from the equilibrium state. Examination of the behaviour of the integral properties of the boundary layer supports this hypothesis. In the present paper a relationship similar to the suggestion of Nash and Goldberg is derived from the local balance of the kinetic energy of the turbulence. Coupling this simple derived relationship to the boundary layer momentum and moment-of-momentum integral equations results in quite accurate predictions of the behaviour of non-equilibrium turbulent boundary layers in arbitrary adverse (given) pressure distributions.

scholarly journals Finite-amplitude steady waves in plane viscous shear flows

1985 ◽  
Vol 160 ◽  
pp. 281-295 ◽  
Author(s):  
F. A. Milinazzo ◽  
P. G. Saffman
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Navier Stokes ◽  
Unidirectional Flow ◽  
Viscous Shear ◽  
Finite Amplitude Waves

Computations of two-dimensional solutions of the Navier–Stokes equations are carried out for finite-amplitude waves on steady unidirectional flow. Several cases are considered. The numerical method employs pseudospectral techniques in the streamwise direction and finite differences on a stretched grid in the transverse direction, with matching to asymptotic solutions when unbounded. Earlier results for Poiseuille flow in a channel are re-obtained, except that attention is drawn to the dependence of the minimum Reynolds number on the physical constraint of constant flux or constant pressure gradient. Attempts to calculate waves in Couette flow by continuation in the velocity of a channel wall fail. The asymptotic suction boundary layer is shown to possess finite-amplitude waves at Reynolds numbers orders of magnitude less than the critical Reynolds number for linear instability. Waves in the Blasius boundary layer and unsteady Rayleigh profile are calculated by employing the artifice of adding a body force to cancel the spatial or temporal growth. The results are verified by comparison with perturbation analysis in the vicinity of the linear-instability critical Reynolds numbers.

Effects of Curvature on Laminar Boundary Layers in Sink-Type Flows

1969 ◽  
Vol 91 (3) ◽  
pp. 353-358 ◽  
Author(s):  
W. A. Gustafson ◽  
I. Pelech
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Momentum Equation ◽  
Similarity Solutions ◽  
Momentum Thickness ◽  
Two Dimensional ◽  
Thickness Increase ◽  
Laminar Boundary Layers ◽  
Converging Channel ◽  
Special Case

The two-dimensional, incompressible laminar boundary layer on a strongly curved wall in a converging channel is investigated for the special case of potential velocity inversely proportional to the distance along the wall. Similarity solutions of the momentum equation are obtained by two different methods and the differences between the methods are discussed. The numerical results show that displacement and momentum thickness increase linearly with curvature while skin friction decreases linearly.

The Isobars in Boundary Layers at Supersonic Speeds

1968 ◽  
Vol 19 (2) ◽  
pp. 105-126 ◽  
Author(s):  
D. F. Myring ◽  
A. D. Young
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Equations Of Motion ◽  
Normal Pressure ◽  
Simple Extension ◽  
Pressure Gradients ◽  
Boundary Layer Flows ◽  
Hyperbolic Flow ◽  
The Individual ◽  
Momentum Integral

SummaryFor boundary layer flows over curved surfaces at moderately high supersonic speeds the existence of normal pressure gradients within the boundary layer becomes important even for small curvatures and they cannot be ignored. The describing equations are basically parabolic in form so that the simplifications inherent in hyperbolic flows would not at first sight seem to be relevant. However, the equations of motion for a two-dimensional, supersonic, rotational, viscous flow are analysed along the lines of a hyperbolic flow and the individual effects of viscosity and vorticity are examined with regard to the isobar distributions. It is found that these two properties have compensating effects and the experimental evidence presented confirms the conclusion that inside the boundary layer the isobars follow much the same rules as those which determine the isobars in the external hyperbolic flow. Since for turbulent boundary layers the fullness of the Mach number profile produces almost linear Mach lines in the boundary layer, this provides a simple extension to the methods of analysis, and the momentum integral equation is reformulated using a swept element bounded by linear isobars. The final equation is similar in form to the conventional one except that the momentum and displacement thicknesses are now defined by integrals along the swept isobars, and all normal pressure gradients due to centrifugal effects are accounted for.

scholarly journals Assessment of inner–outer interactions in the urban boundary layer using a predictive model

2019 ◽  
Vol 875 ◽  
pp. 44-70 ◽  
Author(s):  
Karin Blackman ◽  
Laurent Perret ◽  
Romain Mathis
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Predictive Model ◽  
Boundary Layers ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Limited Range ◽  
Rough Wall ◽  
Wall Boundary ◽  
Layer Flow

Urban-type rough-wall boundary layers developing over staggered cube arrays with plan area packing density, $\unicode[STIX]{x1D706}_{p}$, of 6.25 %, 25 % or 44.4 % have been studied at two Reynolds numbers within a wind tunnel using hot-wire anemometry (HWA). A fixed HWA probe is used to capture the outer-layer flow while a second moving probe is used to capture the inner-layer flow at 13 wall-normal positions between $1.25h$ and $4h$ where $h$ is the height of the roughness elements. The synchronized two-point HWA measurements are used to extract the near-canopy large-scale signal using spectral linear stochastic estimation and a predictive model is calibrated in each of the six measurement configurations. Analysis of the predictive model coefficients demonstrates that the canopy geometry has a significant influence on both the superposition and amplitude modulation. The universal signal, the signal that exists in the absence of any large-scale influence, is also modified as a result of local canopy geometry suggesting that although the nonlinear interactions within urban-type rough-wall boundary layers can be modelled using the predictive model as proposed by Mathis et al. (J. Fluid Mech., vol. 681, 2011, pp. 537–566), the model must be however calibrated for each type of canopy flow regime. The Reynolds number does not significantly affect any of the model coefficients, at least over the limited range of Reynolds numbers studied here. Finally, the predictive model is validated using a prediction of the near-canopy signal at a higher Reynolds number and a prediction using reference signals measured in different canopy geometries to run the model. Statistics up to the fourth order and spectra are accurately reproduced demonstrating the capability of the predictive model in an urban-type rough-wall boundary layer.

Prediction of Reynolds Number Effects on Low-Pressure Turbines Using a High-Order ILES Method

2019 ◽  
Author(s):  
Marc Bolinches-Gisbert ◽  
David Cadrecha Robles ◽  
Roque Corral ◽  
Fernando Gisbert
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Fourth Order ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Low Pressure ◽  
High Order ◽  
Experimental Results ◽  
Numerical Code ◽  
Design System

Abstract A comprehensive comparison between Implicit Large Eddy Simulations (ILES) and experimental results of a modern highlift low-pressure turbine airfoil has been carried out for an array of Reynolds numbers (Re). Experimental data were obtained in a low-speed linear cascade at the Polithecnic University of Madrid using hot-wire anemometry and LDV. The numerical code is fourth order accurate, both in time and space. The spatial discretization of the compressible Navier-Stokes equations is based on a high-order Flux Reconstruction approach while a fourth order Runge-Kutta method is used to march in time the simulations. The losses, pressure coefficient distributions, and boundary layer and wake velocity profiles have been compared for an array of realistic Reynolds numbers. Moreover, boundary layer and wake velocity fluctuations are compared for the first time with experimental results. It is concluded that the accuracy of the numerical results is comparable to that of the experiments, especially for integral quantities such as the losses or exit angle. Turbulent fluctuations in the suction side boundary layer and the wakes are well predicted also. The elapsed time of the is about 140 hours on 40 Graphics Processor Units. The numerical tool is integrated within an industrial design system and reuses pre- and post-processing tools previously developed for another kind of applications. The trend of the losses with the Reynolds number has a sub-critical regime, where the losses scale with Re−1, and a supercrital regime, where the losses scale with Re−1/2. This trend can be seen both, in the simulations and the experiments.

scholarly journals On the decay of dispersive motions in the outer region of rough-wall boundary layers

2019 ◽  
Vol 862 ◽  
Author(s):  
Johan Meyers ◽  
Bharathram Ganapathisubramani ◽  
Raúl Bayoán Cal
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Layers ◽  
Outer Layer ◽  
Stokes Equations ◽  
Outer Region ◽  
Rough Wall ◽  
Rough Boundary ◽  
Mean Flow ◽  
Wall Boundary

In rough-wall boundary layers, wall-parallel non-homogeneous mean-flow solutions exist that lead to so-called dispersive velocity components and dispersive stresses. They play a significant role in the mean-flow momentum balance near the wall, but typically disappear in the outer layer. A theoretical framework is presented to study the decay of dispersive motions in the outer layer. To this end, the problem is formulated in Fourier space, and a set of governing ordinary differential equations per mode in wavenumber space is derived by linearizing the Reynolds-averaged Navier–Stokes equations around a constant background velocity. With further simplifications, analytically tractable solutions are found consisting of linear combinations of $\exp (-kz)$ and $\exp (-Kz)$, with $z$ the wall distance, $k$ the magnitude of the horizontal wavevector $\boldsymbol{k}$, and where $K(\boldsymbol{k},Re)$ is a function of $\boldsymbol{k}$ and the Reynolds number $Re$. Moreover, for $k\rightarrow \infty$ or $k_{1}\rightarrow 0$ (with $k_{1}$ the stream-wise wavenumber), $K\rightarrow k$ is found, in which case solutions consist of a linear combination of $\exp (-kz)$ and $z\exp (-kz)$, and are independent of the Reynolds number. These analytical relations are compared in the limit of $k_{1}=0$ to the rough boundary layer experiments by Vanderwel & Ganapathisubramani (J. Fluid Mech., vol. 774, 2015, R2) and are in reasonable agreement for $\ell _{k}/\unicode[STIX]{x1D6FF}\leqslant 0.5$, with $\unicode[STIX]{x1D6FF}$ the boundary-layer thickness and $\ell _{k}=2\unicode[STIX]{x03C0}/k$.

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