Pohlhausen Method Research Articles

Analytical Study of Laminar Boundary Layers near Blunted Bodies

In high-speed flows, blunt body elements having an irregular shape due to which gas dynamic parameters undergo significant changes are, as a rule, the most thermally loaded parts. In this respect, a quick evaluation of the thermal load on blunt bodies is important. Laminar boundary-layer equations given in special coordinates in a constant axisymmetric flow of a compressible perfect gas are considered. The adhesion condition is accepted as a boundary condition on the wall and it is assumed that the speed and temperature on the boundary correspond to the values of the external flow. The Pohlhausen method introduces the concepts of the displacement thickness and momentum thickness, finds relations between these values and the boundary-layer thickness, and derives a differential equation to determine the boundary-layer form parameter such that all other characteristics of the boundary layer are determined via it. The Pohlhausen method is modified in order to simplify the calculation by excluding the differential equations from it. Similarly to the velocity, a special function including the enthalpy and dimensionless kinetic parameter to be determined is introduced as a biquadratic polynomial. The boundary conditions on the wall and on the border of the boundary layer are used to determine the polynomial coefficients. The kinetic parameter is defined in different ways for bodies of various shapes. We provide the results of applying the proposed method for the calculation of heat flux that have been analyzed numerically (in various papers) in complete systems of Navier-Stokes and Prandtl equations. The comparison of the results shows the efficiency of the proposed method.

Analytical/Numerical Study of Fluid Flow and Heat Transfer Across In-Line Cylinders

An integral approach to boundary-layer analysis is employed to investigate the effects of the longitudinal-pitch ratio on fluid flow and heat transfer from a longitudinal row of circular cylinders immersed in an infinite medium. The momentum equation is solved using the modified von Kármán–Pohlhausen method, which employs a fourth-order velocity profile within the hydrodynamic boundary layer. The potential-flow velocity is obtained by complex potential theory outside the boundary layer. A third-order temperature profile is used in the thermal boundary layer to solve the energy integral equation for the isothermal boundary condition. Closed-form solution is obtained for the heat transfer coefficient for a longitudinal row of circular cylinders immersed in an infinite medium. In the second part of this research activity, a numerical model based on computational fluid dynamics is developed to validate the closed-form solution. The proposed numerical model has been successfully implemented for simulating the flowfield over a longitudinal row of circular cylinders. Numerical simulations have been carried out for various values of the freestream Reynolds number and longitudinal-pitch ratio. The results obtained from the analytical and numerical models have been found to be in good agreement. The maximum percentage error in values of the average Nusselt number obtained from the numerical and analytical solutions is less than 9% and reduces further to a value of less than 4% with an increase in values of the freestream Reynolds number.

An integral treatment of friction during a swash uprush

The effects of bed friction are modelled for the flow near the moving shoreline during the uprush phase of a swash event by extending the Pohlhausen method used by Whitham (1955) to a sloping bed. The tip of the swash near the moving shoreline is treated in an integral sense as a region of uniform velocity, being acted on by the forces of friction, gravity, and the pressure force induced by the frictionless flow behind the swash tip. The bed shear stress is parameterized by using the quadratic dependence on velocity. The theory is compared to data of the shoreline velocity and position in the swash of breaking solitary waves, and the friction coefficient is determined from direct measurements of bed shear stress. The theoretical predictions are in good agreement with the laboratory results in terms of time history of the shoreline velocity and position, as well as the run-up.

Entry flow in curved pipes: turbulent boundary layer approach

ABSTRACTThe development of secondary boundary layer for a steady entry of fluid flow in a horizontal curved pipe is analysed using the turbulent boundary layer approach. Outside the boundary layer (within the central core), the inviscid fluid motion is treated by using the Euler and the continuity equations, whereas inside the boundary layer, the Pohlhausen method is applied, assuming a one-seventh power law of velocity for solving the integrals of the Navier–Stokes equations. The growth of the secondary boundary layer is slow with an increase in azimuthal angle, while it is gradual along the outer pipe-wall with an increase in radial angle and becomes abruptly faster near the separation point. The wall shear stress decreases with azimuthal angle, whereas it increases with radial angle attaining a peak and then decreases to zero at the separation point along the outer pipe-wall.

Analytical and numerical studies on a single-droplet evaporation and combustion under forced convection

Existing droplet evaporation/combustion models in computational fluid dynamics (CFD) simulation of spray combustion are based on simplified 1-D models. Both these models and recently developed 3-D models of single-droplet combustion do not give the conditions for the different existing droplet combustion modes. In this paper, droplet evaporation and combustion are studied both analytically and numerically. In the analytical solution, a 2-D axisymmetric flow surrounding an evaporating and combusting droplet was considered. The governing equations were solved using an integral method, similar to the Karman–Pohlhausen method for solving boundary-layer flows with pressure gradient. The results give a local evaporation rate and flame radius in agreement with experimental results. In numerical simulation, 3-D combusting gas flows surrounding an ethanol droplet were studied. The prediction results show three modes of droplet combustion under different relative velocities, explaining the change in the evaporation constant with an increase in relative velocity observed in experiments. This implies that different droplet combustion models should be developed in simulating spray combustion. The predicted local evaporation rate and flame radius by numerical simulation are in agreement with the analytical solution in the range of azimuthal angles $$0^{\circ }<\theta <90^{\circ }$$ . The numerical results indicate that the drag force of an evaporating and combusting droplet is much smaller than that of a cold solid particle, and thus the currently used drag models should be modified.

Quantifying Compressibility and Slip in Multiparticle Collision (MPC) Flow Through a Local Constriction

The flow of a compressible fluid with slip through a cylinder with an asymmetric local constriction has been considered both numerically, as well as analytically. For the numerical work, a particle-based method whose dynamics is governed by the multiparticle collision (MPC) rule has been used together with a generalized boundary condition that allows for slip at the wall. Since it is well known that an MPC system corresponds to an ideal gas and behaves like a compressible, viscous flow on average, an approximate analytical solution has been derived from the compressible Navier–Stokes equations of motion coupled to an ideal gas equation of state using the Karman–Pohlhausen method. The constriction is assumed to have a polynomial form, and the location of maximum constriction is varied throughout the constricted portion of the cylinder. Results for centerline densities and centerline velocities have been compared for various Reynolds numbers, Mach numbers, wall slip values and flow geometries.

Effects of inertia and surface tension on a power-law fluid flowing down a wavy incline

We consider a thin film of a power-law liquid flowing down an inclined wall with sinusoidal topography. Based on the von Kármán–Pohlhausen method an integral boundary-layer model for the film thickness and the flow rate is derived. This allows us to study the influence of the non-Newtonian properties on the steady free surface deformation. For weakly undulated walls we solve the governing equation analytically by a perturbation approach and find a resonant interaction of the free surface with the wavy bottom. Furthermore, the analytical approximation is validated by numerical simulations. Increasing the steepness of the wall reveals that nonlinear effects like the resonance of higher harmonics grow in importance. We find that shear-thickening flows lead to a decrease while shear thinning flows lead to an amplification of the steady free surface. A linear stability analysis of the steady state shows that the bottom undulation has in most cases a stabilizing influence on the free surface. Shear thickening fluids enhance this effect. The open questions which occurred in the linear analysis are then clarified by a nonlinear stability analysis. Finally, we show the important role of capillarity and discuss its influence on the steady solution and on the stability.

Analysis of hydrodynamics for two-dimensional flow around waving plates

Hydrodynamic characteristics for two-dimensional flow around a waving plate are investigated. Under large Reynolds number approximation, the flow is assumed to be a combination of the outer potential flow and a thin vortex layer, which consists of a boundary layer and a shed free shear layer. A nonlinear mathematical formulation for describing the outer unsteady potential flow coupled with an unsteady boundary layer equation for the inner viscous flow adjacent to the waving plate is developed. A semi-analytical method with a nonlinear Kutta condition imposed at the trailing edge is used to solve the velocity field of the outer flow and the evolution of wake vortex induced by a large-amplitude waving plate. The unsteady boundary layer equation is solved by extending Pohlhausen's method to its unsteady counterpart. The thrust and viscous drag coefficients, propulsive efficiency, and the pattern of wake vortex sheet are discussed.

Steady flows in pipes with finite curvature

Motivated by blood flow in a curved artery, we consider fluid flow through a uniformly curved pipe, driven by a steady pressure gradient. Previous studies have typically considered pipes with asymptotically small curvature in which only the centrifugal effect is retained in the governing equations. Here we consider flows in pipes of finite curvature, and determine the effects of both the centrifugal and Coriolis forces on the flow. The flow behavior is governed by two dimensionless parameters: the curvature, δ, and the Dean number, D. Asymptotic solutions are developed for D⪡1 (using regular perturbation expansions) and D⪢1 (using matched asymptotic expansions and the Pohlhausen method). For intermediate values of D we use a pseudospectral code to obtain solutions. For values of D greater than a critical value, Dc, multiple steady solutions to the governing equations exist; we determine how Dc and the form of the solutions depends on δ. The results indicate that while many of the qualitative features of steady curved-pipe flows are captured by the asymptotic results, the effects of finite curvature can lead to substantial quantitative differences, e.g., in the wall shear stress distribution. The physiological implications of the results for blood flow in arteries are discussed.

Shear stress analysis in axially symmetric flow using the Padé approximation technique

AbstractThis paper evaluates the shear stress on the surface of a circular cylinder in axially symmetric flow within the boundary layer. An extended solution is obtained by established techniques, e.g. asymptotic series or Pohlhausen method. A Padé approximation technique is applied to the extended solution obtained. The region of validity of the solution is extended and the results are then compared and examined against known approximate Pohlhausen and series solutions. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Fluid Flow Around and Heat Transfer From an Infinite Circular Cylinder

AbstractIn this study, an integral approach of the boundary layer analysis is employed to investigate fluid flow around and heat transfer from an infinite circular cylinder. The Von Karman–Pohlhausen method is used to solve momentum integral equation and the energy integral equation is solved for both isothermal and isoflux boundary conditions. A fourth-order velocity profile in the hydrodynamic boundary layer and a third-order temperature profile in the thermal boundary layer are used to solve both integral equations. Closed form expressions are obtained for the drag and the average heat transfer coefficients which can be used for a wide range of Reynolds and Prandtl numbers. The results for both drag and heat transfer coefficients are in good agreement with experimental/numerical data for a circular cylinder.

Shear stress on the surface of a circular cylinder at rest in a moving fluid in axially symmetric flow using the Padé approximation technique

AbstractFollowing the asymptotic series solution of Seban &amp; Bond [1], this paper evaluates the shear stress on the surface of a circular cylinder of radius a at rest in a fluid moving with steady velocity U0 parallel to the axis of the cylinder. The boundary layer builds up from the front of the cylinder and its thickness increases with increasing ξ, where ξ is a non‐dimensional variable representing axial distance from the front of the cylinder. The Seban &amp; Bond solution is valid for small ξ only, near the front of the cylinder where the boundary layer thickness is small compared with a. Curle [2] extended the Seban &amp; Bond results, thus making them valid for larger ξ. Glauert &amp; Lighthill [3] used an asymptotic series to solve the problem in the region where ξ is large. They also used an approximate Pohlhausen method valid for small and large ξ. It is the purpose of this paper to extend the results given by the exact asymptotic series of Seban &amp; Bond so that they are valid for both small and large ξ. This will be done using a Padé approximation technique, which will be compared with the historical results above. It will be seen that the region of validity of the series solution is extended without the need for Curle's extensive analysis and interpolation as in the other cases.

Integral methods for shallow free-surface flows with separation

We study laminar thin film flows with large distortions of the free surface, using the method of averaging across the flow. Two specific problems are studied: the circular hydraulic jump and the flow down an inclined plane. For the circular hydraulic jump our method is able to handle an internal eddy and separated flow. Assuming a variable radial velocity profile as in Kármán–Pohlhausen's method, we obtain a system of two ordinary differential equations for stationary states that can smoothly go through the jump. Solutions of the system are in good agreement with experiments. For the flow down an inclined plane we take a similar approach and derive a simple model in which the velocity profile is not restricted to a parabolic or self-similar form. Two types of solutions with large surface distortions are found: solitary, kink-like propagating fronts, obtained when the flow rate is suddenly changed, and stationary jumps, obtained, for instance, behind a sluice gate. We then include time dependence in the model to study the stability of these waves. This allows us to distinguish between sub- and supercritical flows by calculating dispersion relations for wavelengths of the order of the width of the layer.

Secondary boundary layer and wall shear for fully developed flow in curved pipes

The secondary motion of incompressible fluid for a fully developed turbulent flow in horizontally laid curved pipes is theoretically analysed with the aid of the turbulent boundary–layer approach, integrating the Navier–Stokes equations and satisfying the continuity equation, having mean–time–average parameters. Outside the boundary layer that is developed due to the secondary flow, the inviscid motion of fluid is treated using the Euler and continuity equations. Inside the thin boundary layer, the Pohlhausen method is applied considering a one–seventh power distribution of velocity for solving the integrals of the Navier–Stokes equations. The expression of shear given by Blasius for the turbulent flow in pipes is incorporated into the integrals of the Navier–Stokes equations as an apparent shear to derive an implicit equation that is numerically solved to determine the boundary–layer thickness and subsequently the wall shear stress. The computational results of the boundary–layer thickness and the wall shear stresses are presented in non–dimensional (graphical) form for different Reynolds numbers and curvature ratios. The model is applicable over a wide range of Dean numbers ( D ) 57 ≤ D ≤ × 105.

Revisited results about the theory of steady laminar forced convection for 2D external boundary layer flows

This paper is inspired by the observation in the literature that there was a lack of data about the knowledge of semi-analytical solutions of steady thermal state applied to laminar boundary layer flows in the case of forced convection. Thus, we have made a complete analysis and synthesized the results on both the surface temperature and the surface flux problems, from the development of integral and differential methods, in the range 10 −3 ≤ Pr ≤ 10 2. Moreover, it has been shown that the same ratio, δ T δ = 20 traduces the transition between boundary layer and slug flows, whatever the thermophysical considered case. Finally, we present a warning about the use of 2 and 3-order polynomial shapes by Pohlhausen method which induce a non-unicity of steady state thermal solutions.

LETTERS IN APPLIED AND ENGINEERING SCIENCESTRANSIENT CONJUGATE FREE CONVECTION FROM A VERTICAL FLAT PLATE IN A POROUS MEDIUM SUBJECTED TO A SUDDEN CHANGE IN SURFACE HEAT FLUX

The paper presents a theoretical study using Kármán–Pohlhausen method for describing the transient heat exchange between the boundary-layer free convection and a vertical flat plate embedded in a porous medium. The unsteady behaviour is developed after generation of an impulsive heat flux step at the right-hand side of the plate. Two cases are considered according to whether the plate has a finite thickness or has no thickness. The time and space evolution of the interface temperature is evidenced.

Inertial effects on contact line instability in the coating of a dry inclined plate

It has been observed that when a thin liquid film coats an initially dry inclined plane, a spanwise instability occurs at the leading edge. Here we develop a model for the evolution of this coating film which includes inertia, gravity, surface tension and the contact angle at the leading edge of the film. A Kármán–Pohlhausen method is used to include inertia. We determine steady state profiles of the film and investigate their stability. The predictions of the model are compared to some recent experiments and we find good agreement. This theory gives improvement over a lubrication theory in experiments where Reynolds numbers are significantly larger than one.

Prilog analizi strujanja u graničnom sloju primenom Polhausenove metode

Prilog analizi strujanja u graničnom sloju primenom Polhausenove metode

Free-Surface Boundary-Layer Approach to Study the Bow Vortices Generation Ahead of a Semisubmerged Horizontal Circular Cylinder

In the light of experimental observation of a free-surface shear layer, the flow ahead of a semisubmerged horizontal circular cylinder is modeled as a free-surface boundary layer of concentrated vorticity joining the potential flow below it with the aim to study the generation of bow vortices theoretically. The boundary-layer equations are linearized subject to a suitable assumption and are integrated using basically the Kármán Pohlhausen method. It is found that the free surface moves slower than the layer beneath it, but there is more likelihood of bow vortices being generated by instability of velocity profile rather than by separation and backflow. This is confirmed by the nonmonotonicity in the vorticity profile and the fulfillment of the Görtler instability criterion for flow along curved boundaries, near the body upstream. The position of the point of onset of instability, as stipulated from the above observations, compares well with the position of the free-surface separation point as observed in the experiments.

Bed shear in equilibrium scour around a circular cylinder embedded in a loose bed

The bed shear stress for a horseshoe vortex in the equilibrium scour hole around the base of a circular cylinder, embedded in a loose bed with a uniform approach flow under a clear water regime, is computed from turbulent boundary layer analysis by integrating the Navier—Stokes equations. Inside the thin boundary layer, which is developed by the reverse flow at the bed due to swirl motion, Pohlhausen's method is applied with an arbitrary 1/7th power distribution of velocity, as was assumed by Weber in solving turbulent boundary layer. Outside the boundary layer, the expressions of 3-D flow field given by Dey are used. Finally the above technique produces two unknowns, namely bed shear stress and boundary layer thickness. The expression of wall shear stress given by Blasius for turbulent flow in pipes as well as over flat plates is assumed for solving boundary layer thickness, and subsequently bed shear stress. The computational results of bed shear stress and boundary layer thickness are presented.