Steady flow in a curved tube of triangular cross section

1976 ◽  
Vol 352 (1669) ◽  
pp. 189-211 ◽  
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Steady Flow ◽  
Cross Section ◽  
Numerical Solutions ◽  
Constant Pressure ◽  
Layer Structure ◽  
Asymptotic Model ◽  
Triangular Cross Section ◽  
Curved Tube

The steady flow of a viscous fluid moving under a constant pressure gradient in a curved tube with a uniform triangular cross section is investigated. Numerical solutions of the equations of motion have been found for the range 100-12 000 of the Dean number D = Ga 3 √(2a/ L )/ μv , where G is the constant pressure gradient, a is a dimension of the triangle, L the radius of the circle in which the tube is coiled, μ the viscosity and v the coefficient of kinematic viscosity of the fluid. The results for low D have been checked by an independent numerical method in which the stream function is expanded in a series of powers of D following the method of Dean (1928). All the results have been checked for accuracy by varying the grid size used in the numerical computations. The trend of the results as D increases is examined for evidence of the development of a boundary-layer structure as D → ∞. Some indication is found of the formation of a boundary layer of thickness proportional to D –1/3 near the side walls of the tube with an associated inviscid core region in the centre of the tube. In particular, comparison is made with details of an asymptotic model as D → ∞ proposed by Smith (1976). A measure of agreement with the general characteristics of this model is obtained, although there are some discrepancies in the precise details. It is possible that the range of D considered in the present work is not great enough to form any definite conclusions regarding the precise nature of the flow as D → ∞. A feature of the present results which develops for D > 3000 is that the maximum axial velocity in the tube ceases to occur on the axis of symmetry of the cross section. This feature appears to be generally consistent with numerical results obtained by Cheng & Akiyama (1970) and Hocking (unpublished) for a tube of rectangular cross section. The sequence of corner vortices of the type identified by Moffatt (1964) is found to occur in the numerical solutions. A detailed study of the vortices has already been published (Collins & Dennis 1976).

Compressibility Effects on the Extended Crocco Relation and the Thermal Recovery Factor in Laminar Boundary Layer Flow

2004 ◽  
Vol 126 (1) ◽  
pp. 32-41 ◽  
Author(s):  
B. W. van Oudheusden
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Pressure Gradient ◽  
Boundary Layer Flow ◽  
Numerical Solutions ◽  
Thermal Recovery ◽  
Recovery Factor ◽  
Perturbation Approach ◽  
Layer Flow ◽  
Compressibility Effects

The relation between velocity and enthalpy in steady boundary layer flow is known as the Crocco relation. It describes that for an adiabatic wall the total enthalpy remains constant throughout the boundary layer, when the Prandtl number (Pr) is one, irrespective of pressure gradient and compressibility. A generalization of the Crocco relation for Pr near one is obtained from a perturbation approach. In the case of constant-property flow an analytic expression is found, representing a first-order extension of the standard Crocco relation and confirming the asymptotic validity of the square-root dependence of the recovery factor on Prandtl number. The particular subject of the present study is the effect of compressibility on the extended Crocco relation and, hence, on the thermal recovery in laminar flows. A perturbation analysis for constant Pr reveals two additional mechanisms of compressibility effects in the extended Crocco relation, which are related to the viscosity law and to the pressure gradient. Numerical solutions for (quasi-)self-similar as well as non-similar boundary layers are presented to evaluate these effects quantitatively.

scholarly journals Influence of two-dimensional roughness element on boundary layer structure in the favourable pressure gradient region of the swept wing

2019 ◽  
Vol 234 (1) ◽  
pp. 20-27
Author(s):  
Stepan Tolkachev ◽  
Victor Kozlov ◽  
Valeriya Kaprilevskaya
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Layer Structure ◽  
Roughness Element ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Frequency Range ◽  
Swept Wing ◽  
Boundary Layer Structure ◽  
Roughness Elements

In this article, the results of research about stationary and secondary disturbances development behind the localized and two-dimensional roughness elements are presented. It is shown that the two-dimensional roughness element has a destabilizing effect on the disturbances induced by the three-dimensional roughness element lying upstream. In this case, the two-dimensional roughness element causes the appearance of stationary structures, and then secondary perturbations, whose frequency range lies lower than in the case of the stationary vortices excited by a three-dimensional roughness element.

The structure of two-dimensional separation

1990 ◽  
Vol 220 ◽  
pp. 397-411 ◽  
Author(s):  
Laura L. Pauley ◽  
Parviz Moin ◽  
William C. Reynolds
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Adverse Pressure Gradient ◽  
Navier Stokes ◽  
Stream Velocity ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Shedding Frequency

The separation of a two-dimensional laminar boundary layer under the influence of a suddenly imposed external adverse pressure gradient was studied by time-accurate numerical solutions of the Navier–Stokes equations. It was found that a strong adverse pressure gradient created periodic vortex shedding from the separation. The general features of the time-averaged results were similar to experimental results for laminar separation bubbles. Comparisons were made with the ‘steady’ separation experiments of Gaster (1966). It was found that his ‘bursting’ occurs under the same conditions as our periodic shedding, suggesting that bursting is actually periodic shedding which has been time-averaged. The Strouhal number based on the shedding frequency, local free-stream velocity, and boundary-layer momentum thickness at separation was independent of the Reynolds number and the pressure gradient. A criterion for onset of shedding was established. The shedding frequency was the same as that predicted for the most amplified linear inviscid instability of the separated shear layer.

scholarly journals Application of the Method of Integral Relations to the Calculation of Two-Dimensional Incompressible Turbulent Boundary Layer

1981 ◽  
Vol 48 (4) ◽  
pp. 701-706 ◽  
Author(s):  
W.-S. Yeung ◽  
R.-J. Yang
Keyword(s):  
Experimental Data ◽  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Experimental Results ◽  
Two Dimensional ◽  
Incompressible Turbulent Boundary Layer ◽  
Method Of Integral Relations ◽  
Integral Relations

The orthonormal version of the Method of Integral Relations (MIR) was applied to solve for a two-dimensional incompressible turbulent boundary layer. The flow was assumed to be nonseparating. Flows with favorable, unfavorable, and zero pressure gradient were considered, and comparisons made with available experimental data. In general, the method predicted very well the experimental results for flows with favorable or zero pressure gradient; for flows with unfavorable pressure gradient, it predicted the experimental data well only up to a certain distance from the initial station. This result is due to the flow not being in equilibrium beyond that distance. Finally, the scheme was shown to be efficient in obtaining numerical solutions.

scholarly journals Strongly nonlinear vortices in magnetized ferrofluids

1998 ◽  
Vol 40 (2) ◽  
pp. 146-170 ◽  
Author(s):  
Craig L. Russell ◽  
P.J. Blennerhassett ◽  
P.J. Stiles
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Layer Structure ◽  
Vortex Motion ◽  
Thin Layers ◽  
Marginal Stability ◽  
Weakly Nonlinear ◽  
Boundary Layer Equations ◽  
Conduction State ◽  
Core Boundary

AbstractNonlinear convective roll cells that develop in thin layers of magnetized ferrofluids heated from above are examined in the limit as the wavenumber of the cells becomes large. Weakly nonlinear solutions of the governing equations are extended to solutions that are valid at larger distances above the curves of marginal stability. In this region, a vortex flow develops where the fundamental vortex terms and the correction to the mean are determined simultaneously rather than sequentially. The solution is further extended into the nonlinear region of parameter space where the flow has a core-boundary layer structure characterized by a simple solution in the core and a boundary layer containing all the harmonics of the vortex motion. Numerical solutions of the boundary layer equations are presented and it is shown that the heat transfer across the layer is significantly greater than in the conduction state.

Response of a turbulent boundary layer to sinusoidal free-stream unsteadiness

1990 ◽  
Vol 221 ◽  
pp. 131-159 ◽  
Author(s):  
G. J. Brereton ◽  
W. C. Reynolds ◽  
R. Jayaraman
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Steady Flow ◽  
Boundary Layers ◽  
Free Stream ◽  
Constant Pressure ◽  
External Conditions ◽  
Mean Structure ◽  
The Mean ◽  
Turbulence Production

In this paper, selected findings of a detailed experimental investigation are reported concerning the effects of forced free-stream unsteadiness on a turbulent boundary layer. The forced unsteadiness was sinusoidal and was superimposed locally on an otherwise-steady mainstream, beyond a turbulent boundary layer which had developed under constant-pressure conditions. Within the region over which free-stream unsteadiness was induced, the sinusoidal variation in pressure gradient was between extremes of zero and a positive value, with a positive average level. The local response of the boundary layer to these free-stream effects was studied through simultaneous measurements of the u- and v-components of the velocity fieldAlthough extensive studies of unsteady, turbulent, fully-developed pipe and channel flow have been carried out, the problem of a developing turbulent boundary layer and its response to forced free-stream unsteadiness has received comparatively little attention. The present study is intended to redress this imbalance and, when contrasted with other studies of unsteady turbulent boundary layers, is unique in that: (i) it features an appreciable amplitude of mainstream modulation at a large number of frequencies of forced unsteadiness, (ii) its measurements are both detailed and of high spatial resolution, so that the near-wall behaviour of the flow can be discerned, and (iii) it allows local modulation of the mainstream beyond a turbulent boundary layer which has developed under the well-known conditions of steady, two-dimensional, constant-pressure flowResults are reported which allow comparison of the behaviour of boundary layers under the same mean external conditions, but with different time dependence in their free-stream velocities. These time dependences correspond to: (i) steady flow, (ii) quasi-steadily varying flow, and (iii) unsteady flow at different frequencies of mainstream unsteadiness. Experimental results focus upon the time-averaged nature of the flow; they indicate that the mean structure of the turbulent boundary layer is sufficiently robust that the imposition of free-stream unsteadiness results only in minor differences relative to the mean character of the steady flow, even at frequencies for which the momentary condition of the flow departs substantially from its quasi-steady state. Mean levels of turbulence production are likewise unaffected by free-stream unsteadiness and temporal production of turbulence appears to result only from modulation of the motions which contribute to turbulence production as a time-averaged measure.

The relaxation of a turbulent boundary layer in an adverse pressure gradient

1989 ◽  
Vol 200 ◽  
pp. 367-387 ◽  
Author(s):  
Andrew D. Cutler ◽  
James P. Johnston
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Pressure Gradient ◽  
Eddy Viscosity ◽  
Layer Structure ◽  
Adverse Pressure Gradient ◽  
Great Sensitivity ◽  
A Value ◽  
Gradient Parameter ◽  
Self Similar

The relaxation of a reattached turbulent boundary layer downstream of a wall fence has been investigated. The boundary layer has an adverse pressure gradient imposed upon it which is adjusted in an attempt to bring the boundary layer into equilibrium. This is done by adjusting the pressure gradient so as to bring the Clauser parameter (G) down to a value of about 11.4 and then maintain it constant. In the region from the reattachment point to 2 or 3 reattachment lengths downstream, the boundary layer recovers from the initial major effects of reattachment. Farther downstream (where G is constant) the pressure-gradient parameter changes very slowly and profiles of non-dimensionalized eddy viscosity appear self-similar. However, pressure gradient and eddy viscosity are both roughly twice as large as expected on the basis of previous studies of equilibrium turbulent boundary layers. It is not known whether equilibrium has been achieved in this downstream region. This is another illustration of the great sensitivity of boundary-layer structure to perturbations.

scholarly journals Wavefields forced by long obstacles on a beta-plane

2000 ◽  
Vol 406 ◽  
pp. 221-245 ◽  
Author(s):  
M. A. PAGE ◽  
E. R. JOHNSON
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Layer Structure ◽  
Boundary Layer Equation ◽  
Type Equation ◽  
Ekman Pumping ◽  
Low Friction ◽  
Weakly Nonlinear ◽  
Beta Plane ◽  
Friction Regime

This paper presents analytical and numerical solutions for steady flow past long obstacles on a β-plane. In the oceanographically-relevant limit of small Rossby and Ekman numbers nonlinear advection remains important but viscosity appears only through the influence of Ekman pumping. A reduced boundary-layer-type equation is derived giving the long-obstacle limit of an equation described in Page & Johnson (1990). Analytical solutions are presented or described in various asymptotic limits of this equation and compared with previous results for this or related flows. A novel technique for the numerical solution of the boundary-layer equation, based on a downstream–upstream iteration procedure, is described. Some modifications of the asymptotic layer structure described in Page & Johnson (1991) and Johnson & Page (1993) for the weakly nonlinear low-friction regime are outlined for the case of a lenticular obstacle.

Sign in / Sign up

Export Citation Format

Share Document