Approximate Velocity Profiles Research Articles

Quadratic Polynomial Velocity Profile in Laminar Boundary Layer Thickness

Motion of uid elements can be described by the Navier-Stokes equations. They arise from the use of Law of motion to a uid. In this investigation, two dimensional Navier-Stokes have been engaged. Then they are applied to an incompressible viscous uid movement down an inclined plane with net ow. These leads to examining the effects to the velocity of the motion at various angles of inclination and finding the boundary layer thickness. Viscous laminar incompressible fluid ow also ow on an inclined position which makes it necessary to investigate the ow on an inclined plane. Results that have been achieved are of the ow over horizontal at plate. Solution that has been obtained involves a at photographic film being pulled up by a processing bath by rollers at an angle \(\theta\) to the horizontal. Quadratic polynomial function approximate velocity profile has been obtained under initial boundary layer conditions. This velocity profile has been used in momentum integral equation for ow over an inclined plane to get the boundary layer thickness. Boundary layer thickness is one of the parameters that is used to obtain the ow velocity down inclined plane.

An Experimental Set-Up for the <i>in Vitro</i> Simulation of a Physiological Pulsatile Flow in the Abdominal Aorta

In vitro experimental set-up is an important tool for investigating abdominal aortic aneurysm (AAA). Accurate reproduction of the physiological pulsatile flow waveform in the abdominal aorta at various anatomic locations is an important component of these experimental methods. The objective of this study is to establish an experimental set-up to generate a physiological pulsatile flow for in vitro simulations of the abdominal aorta. The physiological flow was established by a computer-controlled peristaltic pump and the flow field in a circular straight pipe is measured under pulsatile flow conditions by a 2-dimensional particle image velocimetry system (2D-PIV). Experimental results show that the in vitro experimental set-up provides a flow with a period of 2 s, a reasonable cross-sectional velocity distribution and an approximate inlet velocity profile that is close to the human abdominal aorta.

Integral measures of the zero pressure gradient boundary layer over the Reynolds number range ≤Rτ<∞

A recently developed mixing length model of the turbulent shearing stress has been shown to generate a universal velocity profile that provides an accurate approximation to incompressible pipe flow velocity profiles over a wide Reynolds number range [B. J. Cantwell, “A universal velocity profile for smooth wall pipe flow,” J. Fluid Mech. 878, 834–874 (2019)]. More recently, the same profile was shown to accurately approximate velocity profiles in channel flow, the zero pressure gradient boundary layer, and the boundary layer in an adverse pressure gradient [M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows,” AIAA Paper No. 2021-0061, 2021 and M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers,” J. Fluid Mech. (unpublished) (2021)] The universal velocity profile is uniformly valid from the wall to the free stream at all Reynolds numbers from zero to infinity. At a low Reynolds number, the profile approaches the laminar channel/pipe flow limit. The primary measure of the Reynolds number used in this work is the friction Reynolds number Rτ=uτδ/ν. It is a little unusual to use Rτ for the boundary layer since it requires that the velocity profile be cutoff using an arbitrarily defined overall boundary layer thickness, δ. Because of the slow approach of the velocity to the free stream, different conventions used to define the thickness lead to different values of Rτ assigned to a given flow. It will be shown in this paper that, through its connection to channel/pipe flow, the universal velocity profile can be used to define a practically useful, unambiguous, measure of overall boundary layer thickness, called here the equivalent channel half height, δh. For Rτ>≈5000, the universal velocity profile defines a Reynolds number independent shape function that can be used to generate explicit expressions for the infinite Reynolds number behavior of all the usual integral boundary layer measures; displacement thickness, momentum thickness, energy thickness, overall boundary layer thickness, and skin friction. The friction coefficient Cf(Rδ2) generated by the universal velocity profile accurately approximates data over a wide range of momentum thickness Reynolds numbers collected by Nagib et al. [“Can we ever rely on results from wall-bounded turbulent flows without direct measurements of wall shear stress?,” AIAA Paper No. 2004-2392, 2004]. The universal velocity profile is used to integrate the von Kaŕmań boundary layer integral equation [T. von Kármán, “Uber laminaire und turbulente reibung,” Z. Angew. Math. Mech. 1, 233–252 (1921)] in order to generate the various thicknesses and friction velocity as functions of the spatial Reynolds number, Rx=uex/ν.

Influence of cross-sectional velocity profile on flow characteristics of arterial wall modeled as elastic and viscoelastic material.

In the present work, blood flow behavior in a single artery and in arterial network is studied using time domain based one-dimensional wave propagation model retaining the nonlinear convective force. 1-D Navier-Stokes equation is used to model the flow behavior of the blood, using three unknown parameters: flow rate (q), cross-sectional area of artery (A) and pressure (p) based formulation. Three different approximate velocity profile functions across the cross-section namely modified flat, parabolic and the one proposed by Bessems are used to calculate the nonlinear convective force and the frictional force. Two different constitutive models, linear elastic model and standard linear solid model (Zener model) are used to model the arterial wall mechanical behavior. The system of partial differential equations is discretized using finite element and Crank Nicolson methods in space and time domains, respectively. Based on the formulation, an in house finite element code is developed to simulate flow behavior in both a single artery as well as in arterial network consisting of 20 small and large size arteries. Simulations are performed by enforcing a flow rate at the inlet and Windkessel model at the outlet. The results for elastic arterial wall model are found to be in good agreement with the results available in the literature. The flow rate/pressure predictions using different velocity profile functions are found to be nearly the same, however the Bessems velocity profile predicts more closer to 3D results compared to modified flat and parabolic profiles. Whereas, significant difference is found in the results predicted using elastic and viscoelastic artery wall models.

Three-Dimensional Flows in a Rotating Cylinder in the Presence of Turbulent Boundary Layers on End Disks

An analytical model for calculating a viscous incompressible fluid flow in a rotating cylinder with a braking lid and the formation of turbulent boundary layers on the end surfaces is presented. The analysis is made with account for all nonlinear inertial terms in the equations of motion, within the framework of Loitsyanskii's integral relations. The approximate velocity profiles in the boundary layers are preassigned in accordance with the empirical 1/7 law. The main flow is subdivided into an inviscid quasisolid core and a lateral layer, where almost the entire upward part of a circulatory flow is concentrated. The unknown angular velocity of the core and its radial boundary are evaluated from the balance between the moments of the friction forces acting on the main rotating flow and the continuity condition for the circulatory flow.

A New Approach for Modeling Electro-Osmotic Flow Micro-Cooling Systems

The problem of entropy generation due to heat transfer in an electro-osmotic flow with circular type area section micro-channels for small values of electro-kinetic parameterκα(whereκis the Debye length andais the radius of the micro-channel) is investigated analytically. The momentum and energy conservation equations in cylindrical coordinates for an electro-osmotic flow are derived in non-dimensional form. The momentum equation is solved and velocity distribution in terms of modified Bessel function of the first kind is obtained. An approximation is used for the Bessel function of first kind. Considering the approximate velocity profile, the energy equation including viscous dissipation effects is solved to obtain the temperature distribution in terms of parametric values of the electro-kinetic parameter and the Brinkman number. A uniform surface heat flux boundary condition is considered. The importance of this investigation is development of an engineering simple method of design for electro-osmotic circular micro-electronic cooling systems and possible optimizations of these kinds of flows with respect to the second-law of thermodynamics through the micro-coolers.

MOMENTUM INTEGRAL METHOD FOR FORCED CONVECTION IN THERMAL NONEQUILIBRIUM POWER-LAW FLUID-SATURATED POROUS MEDIA

Forced convection heat transfer for power-law fluid flow in porous media was studied analytically. The analytical solutions were obtained based on the Brinkman-extended Darcy model for fluid flow and the two-equation model for forced convection heat transfer. As a closed-form exact velocity profile is unobtainable for the general power-law index, an approximate velocity profile based on the parabolic model is proposed by subscribing to the momentum boundary layer integral method. Heat transfer analysis is based on the two-equation model by considering local thermal nonequilibrium between fluid and solid phases and constant heat flux boundary conditions. The velocity and temperature distributions obtained based on the parabolic model were verified to be reasonably accurate and improvement is justified compared to the linear model. The expression for the overall Nusselt number was derived based on the proposed parabolic model. The effects of the governing parameters of engineering importance such as Darcy number, power-law index, nondimensional interfacial heat transfer coefficient, and effective thermal conductivity ratio on the convective heat transfer characteristics of non-Newtonian fluids in porous media are analyzed and discussed.

A lumped model for blood flow and pressure in the systemic arteries based on an approximate velocity profile function

Previously, by assuming a viscous dominated flow in the boundary layer and an inertia dominated flow in the vessel core, a velocity profile function for a 1D-wave propagation model was derived. Because the time dependent shape of the velocity profile in this boundary layer model depends on the size of the inviscid core and the boundary layer, and thus on the Womersley number, it differs along the arterial tree. In this study we evaluated a lumped model for a vessel segment in which the element configuration is based on physical phenomena described by the boundary layer model and for which all parameters have a physically based quantitative value dependent on the Womersley numbers. The proposed electrical analog consists of a Womersley number dependent resistor and an inductor arranged in parallel, representing the flow impedence in respectively the vessel core and the boundary layer, in series with a second resistor. After incorporating a capacitor representing the vessel compliance in this rigid tube model, the element configuration resembles the configuration of the four-element windkessel model. For arbitrary Womersley numbers the relative impedence of Womersley theory is approximated with high accuracy. In the limits for small and large Womersley numbers the relative impedences of the proposed lumped model correspond exactly to Womersley theory.

A wave propagation model of blood flow in large vessels using an approximate velocity profile function

Lumped-parameter models (zero-dimensional) and wave-propagation models (one-dimensional) for pressure and flow in large vessels, as well as fully three-dimensional fluid–structure interaction models for pressure and velocity, can contribute valuably to answering physiological and patho-physiological questions that arise in the diagnostics and treatment of cardiovascular diseases. Lumped-parameter models are of importance mainly for the modelling of the complete cardiovascular system but provide little detail on local pressure and flow wave phenomena. Fully three-dimensional fluid–structure interaction models consume a large amount of computer time and must be provided with suitable boundary conditions that are often not known. One-dimensional wave-propagation models in the frequency and time domain are well suited to obtaining clinically relevant information on local pressure and flow waves travelling through the arterial system. They can also be used to provide boundary conditions for fully three-dimensional models, provided that they are defined in, or transferred to, the time domain.Most of the one-dimensional wave propagation models in the time domain described in the literature assume velocity profiles and therefore frictional forces to be in phase with the flow, whereas from exact solutions in the frequency domain a phase difference between the flow and the wall shear stress is known to exist. In this study an approximate velocity profile function more suitable for one-dimensional wave propagation is introduced and evaluated. It will be shown that this profile function provides first-order approximations for the wall shear stress and the nonlinear term in the momentum equation, as a function of local flow and pressure gradient in the time domain. The convective term as well as the approximate friction term are compared to their counterparts obtained from Womersley profiles and show good agreement in the complete range of the Womersley parameter α. In the limiting cases, for Womersley parameters α → 0 and α → ∞, they completely coincide. It is shown that in one-dimensional wave propagation, the friction term based on the newly introduced approximate profile function is important when considering pressure and flow wave propagation in intermediate-sized vessels.

Instability of two-phase mixing layers: analysis of exact and approximate base flows from boundary layer theory

The utility of composite error-function velocity profiles for the modeling of gas–liquid shear layers is examined by comparing temporal stability results from such approximate error-function profiles with results from the exact velocity profiles based on the two-fluid boundary layer equations. The stability analysis is restricted to two-dimensional perturbations. The two-fluid boundary layer equations are solved numerically using a shooting method in each fluid layer. The composite error-function profile is constructed by matching displacement thicknesses with the exact solution. With given fluid properties, the displacement thicknesses depend on the asymptotic velocity ratio of the liquid and gas stream in the laboratory frame. For different sets of fluid properties, the maximum growth rates of the Kelvin–Helmholtz instability show good numerical agreement between the exact and the approximate velocity profiles, especially when the asymptotic velocities of the gas and liquid phase are close.

Characterization of the Converging Jet Region in a Triple Torch Plasma Reactor

Enthalpy probe measurements were taken of the converging plasma plume in a triple torch plasma reactor and related to substrate heat flux measurements. Results show excellent entrainment of process gases injected into the converging plasma plume by way of the central injection probe. At lower pressures (≤40 kPa), the plasma volume is equivalent to at least a 3 cm diameter, 4 cm long cylinder, with relatively uniform temperature, velocity, and substrate heat flux profiles when compared to a typical dc arc jet. Converging plasma plume size, substrate heat flux, and enthalpy profiles are also shown to be a strong function of applied system power. Substrate heat flux measurements show smaller radial gradients than enthalpy probe measurements, because of the high radial velocity component of gases above the substrate boundary layer. Enthalpy probe measurements were also conducted for diamond deposition conditions and approximate temperature and velocity profiles obtained. Problems with the uniform gas mixture assumption prohibited more accurate measurements. Reproducibility of enthalpy measurement results was shown with an average standard deviation of 11.8% for the velocity and 7.6% for the temperature measurements.

Boundary-layer analysis of the thermal bar

The thermal bar, a descending plane plume of fluid at the temperature of maximum density (3.98° C in water), is analysed as a laminar free-convection boundary layer, following the example of Kuiken & Rotem (1971) for the plume above a line source of heat. Numerical integration of the similarity form of the boundary-layer equations yields values of the vertical velocity and temperature gradient on the centre line and the horizontal velocity induced outside the thermal bar as functions of Prandtl number σ. The asymptotic behaviour of these parameters for both large and small σ is also obtained; in these cases, the thermal bar has a two-layer structure, and the method of matched asymptotic expansions is used. For the intermediate case σ= 1, an analytical calculation using approximate velocity and temperature profiles in the integrated boundary-layer equations yields good agreement with the numerical results. The applicability of the results to naturally occurring thermal bars (e. g. in lakes) is limited, but the laminar-flow analysis is likely to relate more closely to the phenomenon on a laboratory scale.

On the use of approximate velocity profiles for modeling multiphase reactors

AbstractThe use of an approximate axial gas velocity profile in a two‐phase reactor model has been studied. The approximate velocity was assumed to vary linearly with conversion, which reduces the number of differential equations to be solved. The approximate model was studied in systems with one or two reactions. In most cases, the approximate and rigorous solutions were in good agreement. However, when mass transfer strongly influenced the velocity profile, errors in conversion became significant (∼ 15%). Reactions with no change in the number of moles are also subject to errors if the rigorous velocity profile is not accounted for.

Laminar forced convection in converging or diverging planar symmetric ducts

Approximate velocity profiles are developed and a new transformation is introduced to reduce the heat transfer problem for laminar forced convection inside a diverging or converging planar symmetric duct subjected to uniform wall temperature to the standard thermal entrance region problem of Graetz type for flow between two parallel plates.

Velocity profiles and form of a laminar jet in immiscible liquid-liquid systems

An exact numerical solution for the velocity profiles and the form of a laminar jet in immiscible Newtonian liquid—liquid systems is obtained. The main difficulty connected with the simultaneous integration of the equations of motion of the jet and of the continuous phase in the presence of an unknown interface is overcome by using an indivisible finite-difference scheme. This avoids any additional iteration for defining the unknown velocities of the points of the interface. The solution of the equations of motion as written in a boundary layer approximation is a function of the following non-dimensional numbers: the Reynolds numbers of each phase, the Weber and Froude numbers and the ratio of the densities. The influence of some of these parameters on the jet behaviour is illustrated as well as the influence of the initial velocity profiles at the nozzle exit. A comparison is made with some known results. Important differences are found to exist between the exact and the approximate velocity profiles and their gradients at the interface. It seems that these differences result from the comparatively inexact description of the boundary layer of the continuous phase when using moment methods. Such a conclusion limits the applicability of the approximate moment solutions to heat and mass transfer problems as well as to the jet stability analysis.

Theoretical analysis on the flow of melts in a curved circular tube

AbstractA theoretical study is made of molten polymer flow in a curved circular tube. To obtain the velocity profile formed in the tube, a variational approach is used. A variational equation for viscous dissipation has been so determined that its Euler equation satisfies the equation of motion. An approximate velocity profile which satisfies the prescribed boundary conditions is assumed as a function involving unknown coefficients and is substituted into the variational equation to determine the unknown coefficients. Various velocity profiles of polymer melts for different flow behavior indices have been obtained under the assumption that polymer melts obey a power law.

The Discharge Coefficient of Spool Type Hydraulic Valve : 3rd Report, The Taper Land Spool Valve

In this report, the discharge property of a taper land spool valve is analyzed theoretically on the assumption of laminar flow and verified experimentally. Theoretically, we assume a taper land channel as a two-dimensional divergent or convergent channel, and obtain a pressure loss with approximate velocity profile. In the inlet part, we apply the similarity of velocity profile to a boundary layer equation and obtain a pressure loss in the inlet length. Experimentally, we make a model of two-ports valve and obtain precise data. Taper angle is set to be 1.5°∼6°, spool displacement 0.5∼4 mm, Reynolds number 10∼1, 200. In this range we experiment, and compute the discharge property in the same range. In the result, experiment data show good coincidence with computation result. Further, the influences of back pressure of the valve are also discussed.

Improved Linearized Velocity Profiles for Turbulent Free Shear Layers

The longitudinal motion equation for turbulent boundary layers is linearized in a manner which retains the spirit of Reichardt’s inductive theory. The resulting approximate velocity profiles for isobaric free shear layers are found to be functions of the density field as well as the velocity of the secondary stream. Using the amount of transverse shift, which is required to satisfy the momentum integral equation, as a figure of merit, it is shown that the alternate profiles are generally better representations of the flow field than distributions obtained with the classic Oseen linearization for both developing and fully developed free layers. The alternate velocity distributions, in combination with integral conservation equations, are shown to yield characteristic velocities in the midpart of the profile which are in agreement with those obtained from more elaborate nonlinear analyses.

Hypersonic laminar separation on a cone with slip flow

This is an experimental investigation of hypersonic, axisymmetric, laminar separation of a thick, slip-flow viscous layer on a cone that is isothermal and slightly cooled. Minutely blunted cones with half-angles of 5° and attached compression flares from 20° to 90° were used. The freestream had a Mach number of about 14.5 at the model tip and was slightly divergent. Experimental observations included surface-pressure measurements and Pilot surveys of attached and separated flows. These measurements are converted into approximate velocity profiles, Mach number profiles, displacement thicknesses, and shock shapes. The flows just upstream of separation interaction were influenced by 1) tip bluntness as far as 1000 tip diameters downstream of the tip, 2) weak interaction, and 3) the slight divergence of the freestream. Nevertheless, surface pressures and approximate displacement thicknesses for the separation-inter action regions on the cones are correlated by hypersonic, freeinteraction, similarity variables that are obtained from a Chapman analysis.

Hypersonic laminar boundary layer approaching the base of a slender body.

An asymptotic description is obtained for the acceleration of a laminar, hypersonic boundary layer approaching a sharp corner, assuming small interaction with the outer inviscid flow. Viscous forces are neglected except in a thin sublayer. The initial part of the expansion takes place over a distance O(Mj), where Me is the external Mach number and d is the boundarylayer thickness. Here the transverse pressure gradient is small and a solution can be obtained analytically. Within a distance O(S) from the corner, the effect of streamline curvature is essential, and a numerical solution is obtained by the method of integral relations for a single strip. The solution for surface pressure is compared with experimental results and with a similar theory for a laminar, supersonic boundary layer approaching a corner, and an approximate velocity profile at the corner is calculated.