Local Reynolds Number Research Articles

Thermal entry length in falling liquid films at high Reynolds numbers

A behavior of the length of thermal entry was studied experimentally in the vertically falling wavy liquid film at high Reynolds numbers with application of the field methods for thickness and temperature measurements. According to data, obtained by a high-speed IR camera, the zones, whose temperatures exceed the inlet temperature of the film, are formed at 7–10mm from upper edge of the heater in the interrivulet area between the crests of developed 3D synchronous waves even at low heat fluxes. It was determined that in the area of rivulets, formed on the crests of 3D synchronous waves, the length of thermal entry is higher, but it decreases drastically with a rise of heat flux. Analysis of data on the film thickness, obtained by the fluorescent method, has shown that between the synchronous 3D waves in spanwise direction the average film thickness is lower. Respectively, the values of local Reynolds numbers of the film and distances before the appearance of temperature nonuniformity decrease. In these regions the effect of thermocapillary forces is obvious even at relatively low heat fluxes, what finally leads to a considerable decrease in the length of thermal entry.

Intermittency and local Reynolds number in Navier-Stokes turbulence: A cross-over scale in the Caffarelli-Kohn-Nirenberg integral

We study space-time integrals, which appear in the Caffarelli-Kohn-Nirenberg (CKN) theory for the Navier-Stokes equations analytically and numerically. The key quantity is written in standard notations \documentclass[12pt]{minimal}\begin{document}$\delta (r)=1/(\nu r)\int _{Q_r}\left|\nabla {\bm u}\right|^2\,d{{\bm x}} \,dt$\end{document}δ(r)=1/(νr)∫Qr∇u2dxdt, which can be regarded as a local Reynolds number over a parabolic cylinder Qr. First, by re-examining the CKN integral, we identify a cross-over scale \documentclass[12pt]{minimal}\begin{document}$r_* \propto L\left( \frac{ \overline{\Vert \nabla \bm {u} \Vert ^2_{L^2}} }{\Vert \nabla \bm {u} \Vert ^2_{L^\infty }} \right)^{1/3},$\end{document}r*∝L‖∇u‖L22¯‖∇u‖L∞21/3, at which the CKN Reynolds number δ(r) changes its scaling behavior. This reproduces a result on the minimum scale rmin in turbulence: \documentclass[12pt]{minimal}\begin{document}$r_{\rm min}^2 \Vert \nabla {\bm u}\Vert _\infty \propto \nu ,$\end{document}r min 2‖∇u‖∞∝ν, consistent with a result of Henshaw et al. [“On the smallest scale for the incompressible Navier-Stokes equations,” Theor. Comput. Fluid Dyn. 1, 65 (1989)10.1007/BF00272138]. For the energy spectrum E(k) ∝ k−q (1 < q < 3), we show that r* ∝ νa with \documentclass[12pt]{minimal}\begin{document}$a=\frac{4}{3(3-q)}-1$\end{document}a=43(3−q)−1. Parametric representations are then obtained as \documentclass[12pt]{minimal}\begin{document}$\Vert \nabla {\bm u}\Vert _\infty \propto \nu ^{-(1+3a)/2}$\end{document}‖∇u‖∞∝ν−(1+3a)/2 and rmin ∝ ν3(a+1)/4. By the assumptions of the regularity and finite energy dissipation rate in the inviscid limit, we derive \documentclass[12pt]{minimal}\begin{document}$\lim _{p \rightarrow \infty }\frac{\zeta _p}{p}=1 - \zeta _2$\end{document}limp→∞ζpp=1−ζ2 for any phenomenological models on intermittency, where ζp is the exponent of pth order (longitudinal) velocity structure function. It follows that ζp ⩽ (1 − ζ2)(p − 3) + 1 for any p ⩾ 3 without invoking fractal energy cascade. Second, we determine the scaling behavior of δ(r) in direct numerical simulations of the Navier-Stokes equations. In isotropic turbulence around Rλ ≈ 100 starting from random initial conditions, we have found that δ(r) ∝ r4throughout the inertial range. This can be explained by the smallness of a ≈ 0.26,with a result that r* is in the energy-containing range. If the β-model is perfectly correct, the intermittency parameter a must be related to the dissipation correlation exponent μ as \documentclass[12pt]{minimal}\begin{document}$\mu =\frac{4a}{1+a} \approx 0.8,$\end{document}μ=4a1+a≈0.8, which is larger than the observed μ ≈ 0.20. Furthermore, corresponding integrals are studied using the Burgers vortex and the Burgers equation. In those single-scale phenomena, the cross-over scale lies in the dissipative range. The scale r* offers a practical method of quantifying intermittency. This paper also sorts out a number of existing mathematical bounds and phenomenological models on the basis of the CKN Reynolds number.

Studies on Ignition Delay and Flame Spread in High-Performance Solid Rocket Motors

Accurate prediction of ignition delay and flame spread rate in solid propellant rocket motors is of great topical interest. In this paper using a standard k-ω turbulence model numerical studies have been carried out to examine the influence of solid rockets port geometry on ignition delay and the flame spread pattern. We observed that with the same inflow conditions and propellant properties heat flux histories and ignition time sequence are different for different port geometries. We conjectured from the numerical results that in solid rocket motors with highly loaded propellants, mass flux of the hot gases moving past the burning surface is large. Under these conditions, the convective flux to the surface of the propellant will be enhanced, which in turn enhance the local Reynolds number. This amounts a reduction in heat transfer film thickness and enhanced heat transfer to the propellant with consequent enhancement in the dynamic burn rate resulting the undesirable starting pressure transient. We concluded that, the more accurate description of gas phase to surface heat transfer process will give a better prediction and control of ignition delay and flame spread rate in solid propellant rockets.

Transition Modeling for Vortex Generating Jets on Low-Pressure Turbine Profiles

Steady blowing vortex generating jets (VGJ) on highly-loaded low-pressure turbine profiles have shown to be a promising way to decrease total pressure losses at low Reynolds-numbers by reducing laminar separation. In the present paper, the state of the art turbomachinery design code TRACE with RANS turbulence closure and coupled γ-ReΘ transition model is applied to the prediction of typical aerodynamic design parameters of various VGJ configurations in steady simulations. High-speed cascade wind tunnel experiments for a wide range of Reynolds-numbers, two VGJ positions, and three jet blowing ratios are used for validation. Since the original transition model overpredicts separation and losses at Re2is≤100·103, an extra mode for VGJ induced transition is introduced. Whereas the criterion for transition is modeled by a filtered Q vortex criterion the transition development itself is modeled by a reduction of the local transition-onset momentum-thickness Reynolds number. The new model significantly improves the quality of the computational results by capturing the corresponding local transition process in a physically reasonable way. This is shown to yield an improved quantitative prediction of surface pressure distributions and total pressure losses.

Görtler instability of the axisymmetric stagnation-point flow along a concave wall

Axisymmetric stagnation-point flow, if it is along a concave wall, is susceptible to the Görtler instability, which is governed by partial differential equations with respect to both the normal-to-wall coordinate and the local Reynolds number. A series solution of the exact disturbance equations is obtained in the form expanded into inverse powers of the Reynolds number. The lowest-order equations concerning the leading terms of the series construct a non-parallel version of Görtler's eigenvalue problem and present a neutral stability curve with the critical point in a finite range of wave numbers for disturbances of the steady longitudinal-vortex type. Higher-order terms of the series solution indicate the dependence of the Görtler instability on the local Reynolds number and suggest a quantitative limit of validity of the Görtler approximation based on the lowest-order equation system alone. In the case of the concave wall that slightly rotates around the axis of symmetry, the instability induces oblique traveling waves instead of steady longitudinal vortices.

Numerical investigation of single phase and two phase flow through thin orifices in horizontal pipes

Pressure drop due to single phase flow of water and two phase flow of air-water mixture through thin orifices in horizontal pipes have been numerically investigated. Two-phase computational fluid dynamics (CFD) calculations, using Eulerian–Eulerian model have been employed to calculate the pressure drop through orifices. The operating condition covers the gas and liquid superficial velocity ranges Vsg=0.3–4 m/s and Vsl=0.6–2 m/s, respectively. The local pressure drops have been obtained by means of extrapolation from the computed upstream and downstream linearized pressure profiles to the orifice section. Simulations for the single-phase flow of water have been carried out for local liquid Reynolds number (Re based on orifice diameter) ranging from 3×104 to 2×105 to obtain the discharge coefficient and the two-phase local multiplier, which when multiplied with the pressure drop of water (for same mass flow of water and two phase mixture) will reproduce the pressure drop for two phase flow through the orifice. The effect of orifice geometry on two-phase pressure losses has been considered by selecting eight different orifice plates with two area ratios (σ=0.73 and σ=0.54) and four different thicknesses (s/d = 0.025-0.59). The results obtained from numerical simulations are validated against experimental data from the literature and are found to be in good agreement. Keywords: Orifice, Pressure drop, Two phase flow, Computational fluid dynamics, Area ratio, Discharge coefficient

Leray-α Regularization of the Smagorinsky-Closed Filtered Equations for Turbulent Jets at High Reynolds Numbers

The article reports on blending of the Leray-α regularization with the conventional Smagorinsky subgrid-scale closure as an option for large-eddy-simulation of turbulent flows at very high Reynolds number on coarse meshes. The model has been tested in the self-similar far-field region of a jet at a range of Reynolds numbers spanning over two decades (4×103, 4×104 and 4×105) on two very coarse meshes of 2×105 and 3×104 mesh cells. The results are compared with the well-resolved DNS for \(Re_D=4\times 10^3\) on 15 million cells and experimental data for higher Re numbers. While the pure Leray-α can fail badly at high Re numbers on very coarse meshes, a blending of the two strategies by adding a small amount of extra-dissipation performs well even at a huge jet Reynolds number of \(Re_D=4\times 10^5\) on a very coarse mesh (2×105 cells), despite the ratio of the typical mesh spacing to the Kolmogorov length exceeding 300. It is found that the main prerequisite for successful LES, both for the classic Smagorinsky and the blended Leray-α/Smagorinsky model, is to resolve the shear-length \(L_s=\sqrt{\varepsilon/{\cal S}^3}\) (where \({\cal S}\) is the shear-rate modulus), defined by the constraint Δ/Ls < 1, where Δ is the typical mesh-cell size. For the mixed Leray-α/Smagorinsky model the regularization parameter should also be related to the shear-length rather than the local mesh size or Reynolds number, for which we propose a guide criterion α = 0.15÷0.3 Ls.

Entropy generation in hydrodynamic slip flow over a vertical plate with convective boundary

The present article aims to report the effects of hydrodynamic slip on entropy generation in the boundary layer flow over a vertical sur- face with convective boundary condition. Suitable similarity transformations are used to transform the fundamental equations of hydro- dynamic and thermal boundary layer flow into ordinary differential equations. The governing equations are then solved numerically us- ing the shooting method and the velocity and the temperature profiles are obtained for various values of parameters involved in the gov- erning equations. The expressions for the entropy generation number and the Bejan number are presented and the results are discussed graphically and quantitatively for the slip parameter, the local Grashof number, the Prandtl number, the local convective heat transfer parameter, the group parameter and the local Reynolds number. It is observed that due to the presence of slip, entropy production in a thermal system can be controlled and reduced.

Single-Phase and Two-Phase Flow Through Thin and Thick Orifices in Horizontal Pipes

Two-phase flow pressure drops through thin and thick orifices have been numerically investigated with air–water flows in horizontal pipes. Two-phase computational fluid dynamics (CFD) calculations, using the Eulerian–Eulerian model have been employed to calculate the pressure drop through orifices. The operating conditions cover the gas and liquid superficial velocity ranges Vsg = 0.3–4 m/s and Vsl = 0.6–2 m/s, respectively. The local pressure drops have been obtained by means of extrapolation from the computed upstream and downstream linearized pressure profiles to the orifice section. Simulations for the single-phase flow of water have been carried out for local liquid Reynolds number (Re based on orifice diameter) ranging from 3 × 104 to 2 × 105 to obtain the discharge coefficient and the two-phase local multiplier, which when multiplied with the pressure drop of water (for same mass flow of water and two phase mixture) will reproduce the pressure drop for two phase flow through the orifice. The effect of orifice geometry on two-phase pressure losses has been considered by selecting two pipes of 60 mm and 40 mm inner diameter and eight different orifice plates (for each pipe) with two area ratios (σ = 0.73 and σ = 0.54) and four different thicknesses (s/d = 0.025–0.59). The results obtained from numerical simulations are validated against experimental data from the literature and are found to be in good agreement.

Laminar-turbulent cycles in inclined lock-exchange flows

We consider strongly confined, stably stratified shear flows generated as a lock exchange in a tube inclined at an angle of θ=45(∘). This paper focuses on a transitional regime, in which the flow alternates between two distinct states: laminar, parallel shear flow and intense transverse motion characteristic of turbulence. Laminar-turbulent cycles were captured at Atwood numbers At≡(ρ(2)-ρ(1))/(ρ(1)+ρ(2)) ranging from 2.45×10(-3) to 4.0×10(-3), where (ρ(1),ρ(2)) are the initial densities of the two fluids, with multiple cycles observed at At=2.55×10(-3). The evolution of the density and velocity fields in these flows was measured simultaneously using laser-induced fluorescence and particle image velocimetry. During each laminar-turbulent cycle, the axial velocity exhibits a distinctive ramp-cliff pattern, indicating that the flow accelerates as it relaminarizes, then decelerates rapidly as the Kelvin-Helmholtz billows break down. Within the range of experimental conditions, transverse stratification does not directly determine the onset of instability. Instead, the data suggest that a necessary criterion for the onset of instability is for the local Reynolds number to exceed 2200, with only a weak dependence on the Richardson number.

Thermal entry length in a falling liquid film at high reynolds numbers

Using the fluorescence and the infrared technique of thickness and temperature measuring, experimental study is performed on variation of the thermal entry length in a falling water film at high Reynolds numbers. The data obtained by means of a high-speed infrared camera show that, in the interrivulet area occurring between the crests of the developed synchronous 3D waves, even at low heat fluxes, at distances of 7–10 mm from the upper heater edge, zones are formed with the temperature above the initial film temperature. The thermal entry length is shown to be larger within the area of the rivulets forming over the synchronous 3D wave crests, but it decreases rapidly with the heat flux density increase. Analysis of the film thickness data obtained by the fluorescent technique shows that, within the areas between the synchronous 3D waves, the averaged film thickness is less. Respectively, the values of the local film Reynolds number decrease, as do the distances to the localities of the temperature non-uniformities. In those places, even at relatively low heat fluxes, the influence of the thermo-capillary force takes place thus causing a significant decrease in the thermal entry length.

First instability of the flow of shear-thinning and shear-thickening fluids past a circular cylinder

AbstractThe first bifurcation and the instability mechanisms of shear-thinning and shear-thickening fluids flowing past a circular cylinder are studied using linear theory and numerical simulations. Structural sensitivity analysis based on the idea of a ‘wavemaker’ is performed to identify the core of the instability. The shear-dependent viscosity is modelled by the Carreau model where the rheological parameters, i.e. the power-index and the material time constant, are chosen in the range $0. 4\leq n\leq 1. 75$ and $0. 1\leq \lambda \leq 100$. We show how shear-thinning/shear-thickening effects destabilize/stabilize the flow dramatically when scaling the problem with the reference zero-shear-rate viscosity. These variations are explained by modifications of the steady base flow due to the shear-dependent viscosity; the instability mechanisms are only slightly changed. The characteristics of the base flow, drag coefficient and size of recirculation bubble are presented to assess shear-thinning effects. We demonstrate that at critical conditions the local Reynolds number in the core of the instability is around 50 as for Newtonian fluids. The perturbation kinetic energy budget is also considered to examine the physical mechanism of the instability.

SUPG and discontinuity-capturing methods for coupled fluid mechanics and electrochemical transport problems

Electrophoresis is the motion of charged particles relative to the surrounding liquid under the influence of an external electric field. This electrochemical transport process is used in many scientific and technological areas to separate chemical species. Modeling and simulation of electrophoretic transport enables a better understanding of the physicochemical processes developed during the electrophoretic separations and the optimization of various parameters of the electrophoresis devices and their performance. Electrophoretic transport is a multiphysics and multiscale problem. Mass transport, fluid mechanics, electric problems, and their interactions have to be solved in domains with length scales ranging from nanometers to centimeters. We use a finite element method for the computations. Without proper numerical stabilization, computation of coupled fluid mechanics, electrophoretic transport, and electric problems would suffer from spurious oscillations that are related to the high values of the local Peclet and Reynolds numbers and the nonzero divergence of the migration field. To overcome these computational challenges, we propose a stabilized finite element method based on the Streamline-Upwind/Petrov-Galerkin (SUPG) formulation and discontinuity-capturing techniques. To demonstrate the effectiveness of the stabilized formulation, we present test computations with 1D, 2D, and 3D electrophoretic transport problems of technological interest.

High Intensity Drying in Porous Materials

High intensity drying is used to characterize those situations for which the drying medium is sufficiently above the saturation temperature of water to preclude the existence of a two-phase zone. In the present work, three models are applied to high intensity drying of porous materials. The three models are: (1) a traditional one-equation model that assumes local thermal equilibrium (LTE); (2) a two-equation model that utilizes lumped capacitance assumption to predict the heat transfer to the solid phase; and (3) a two-equation model that utilizes a more precise quasi-analytical approach to more accurately characterize the conduction in the solid phase. In addition, the relationship between pressure and the drying conditions and material properties is examined since elevated pressure that can occur during high intensity drying is potentially destructive. An implicit finite difference scheme is utilized to determine the drying rate in a porous medium saturated with water and undergoing the phase change process. The case for low local Reynolds number is considered, hence Nusselt number is assumed constant. Results illustrate that the one-equation model does not yield accurate results when the thermophysical properties characterized by the volume weighted ratio of thermal diffusivities, C &gt; 10 (within 5% error). Hence, a two-equation model is suggested. In addition, consistent with the established “rule of thumb,” for Biot number, Biv &lt; 0.1, the traditional two-equation model which makes the lumped capacitance assumption for the solid phase compares well with a two-equation model that more accurately predicts the time dependent diffusion in the solid phase using Duhamel’s theorem. The peak pressures observed during drying for a range of Darcy number and surface heat flux are presented.

Numerical Investigation of Single Phase and Two Phase Flow through Thin Orifices in Horizontal Pipes

Pressure drop due to single phase flow of water and two phase flow of air-water mixture through thin orifices in horizontal pipes have been numerically investigated. Two-phase computational fluid dynamics (CFD) calculations, using Eulerian-Eulerian model have been employed to calculate the pressure drop through orifices. The operating condition covers the gas and liquid superficial velocity ranges V sg =0.3-4 m/s and V sl =0.6-2 m/s, respectively. The local pressure drops have been obtained by means of extrapolation from the computed upstream and downstream linearized pressure profiles to the orifice section. Simulations for the single-phase flow of water have been carried out for local liquid Reynolds number (Re based on orifice diameter) ranging from 3×10 4 to 2×10 5 to obtain the discharge coefficient and the two-phase local multiplier, which when multiplied with the pressure drop of water (for same mass flow of water and two phase mixture) will reproduce the pressure drop for two phase flow through the orifice. The effect of orifice geometry on two-phase pressure losses has been considered by selecting eight different orifice plates with two area ratios (σ=0.73 and σ=0.54) and four different thicknesses (s/d = 0.025-0.59). The results obtained from numerical simulations are validated against experimental data from the literature and are found to be in good agreement.

A GENERALIZED DIFFERENTIAL TRANSFORM METHOD FOR COMBINED FREE AND FORCED CONVECTION FLOW ABOUT INCLINED SURFACES IN POROUS MEDIA

In this article, we apply the differential transform method (DTM) to obtain approximate analytical solutions of combined free and forced (mixed) convection about inclined surfaces (or wedges) in a saturated porous medium. Both aiding and opposing flows are considered. It is found that the parameter mixed convection from inclined surfaces in porous media is Gr/Re, where Gr is the local Grashof number and Re is the local Reynolds number. DTM solutions are obtained for mixed convection from an isothermal vertical flat plate as well as an inclined plate with constant heat flux having an inclination of 45°. Temperature and velocity profiles for these two cases at different values of Gr/Re are presented. The similarity transformations are applied to reduce the governing partial differential equations (PDEs) to a set of nonlinear coupled ordinary differential equations (ODEs) in dimensionless form. DTM is used to solve the nonlinear differential equations governing the problem in the form of series with easily computable terms. Thereafter a Padé approximant is applied to the solutions to increase the convergence of the given series. Excellent correlation between DTM-Padé and numerical quadrature (shooting) solutions is achieved. The DTM-Padé simulation is shown to be a robust benchmarking tool providing an excellent means of validation of numerical methods. The study has applications in geothermal energy systems, chemical engineering filtration systems, and packed beds.

Using bulk convection in a microtensiometer to approach kinetic-limited surfactant dynamics at fluid–fluid interfaces

The impact of transport of surfactants to fluid–fluid interfaces is complex to assess and model, as many processes are in the regime where kinetics, diffusion and convection are comparable. Using the principle that the timescale for diffusion decreases with increasing curvature, we previously developed a microtensiometer to accurately measure fundamental transport coefficients via dynamic surface tension at spherical microscale liquid–fluid interfaces. In the present study, we use a low Reynolds number flow in the bulk solution to further increase the rate of diffusion. Dynamic surface tension is measured as a function of Peclet number and the results are compared with a simplified convection–diffusion model. Although a transition from diffusion to kinetic-limited transport is not observed experimentally for the surfactants considered, lower bounds on the adsorption and desorption rate constants are determined that are much larger than previously reported rate constants. The results show that the details of the flow field do not need to be controlled as long as the local Reynolds number is low. Aside from other pragmatic advantages, this experimental tool and analysis allows the governing mechanisms of surfactant transport at liquid–fluid interfaces to be quantified using flow near the interface to decrease the length scale for diffusion, separating the relevant timescales.

High-temperature gradient effect on a turbulent channel flow using thermal large-eddy simulation in physical and spectral spaces

This paper studied nonisothermal plane channel flows using the method of thermal large-eddy simulations (TLES). Several temperature ratios (TR =TH /TC ) were investigated, where TC and TH are the temperatures of the cold and hot sides, respectively. Each TR case is further considered for two wall Reynolds numbers (Re τm ), 180 and 395, where the wall Reynolds number was defined as the average of the local wall Reynolds number values obtained for the cold and hot sides, Re τC and Re τH . For a given wall Reynolds number Re τm , it was demonstrated that the increased temperature ratio directly led to an enhanced disparity between the local wall Reynolds numbers of the cold and hot sides. Essentially, the wall Reynolds number for the cold side (Re τC ) is increased, but its companion value for the hot side (Re τH ) is decreased. In the case of large temperature ratio, the asymmetry of the mean velocity and temperature profiles is also increased. The major findings are that for high-temperature ratios, the flow on the hot side could relaminarize, as indicated by the parabolicity of the axial flow profile on the hot side; however, this process can be overcome if the flow has a greater turbulent intensity, as illustrated for the higher-Reynolds-number case studied in the paper. Finally, the impact of the temperature gradient on the turbulent fluctuations is investigated by controlling the influence of the Reynolds number. Four nonisothermal energy spectra on the cold and the hot side are compared and contrasted with eight isothermal flows. Here, the isothermal flows were chosen so that their Reynolds numbers have the same values as those Re τC and Re τH previously evaluated from the nonisothermal flows. The isothermal spectra reproduced the classical Kolmogorov spectra when the Reynolds numbers were sufficiently high. As a result, any observed deviation of the spectra index of the nonisothermal spectra from that of the isothermal one is a direct consequence of the temperature gradient.

Dynamos in rotating compressible convection

Motivated by open questions in fundamental dynamo theory, the overall aim of this paper is to investigate some of the properties of dynamo action in rotating compressible convection. We study dynamo action in a convective layer of electrically-conducting, compressible fluid, rotating about the vertical axis. In order to identify the effects of rotation, we also carry out an equivalent set of calculations of convectively-driven dynamo action in a non-rotating layer. Whether or not the layer is rotating, the convection acts as a small-scale dynamo provided that the magnetic diffusivity is small enough. Defining the magnetic Reynolds number in terms of the horizontal scales of motion, we find that rotation reduces the critical value of this parameter above which dynamo action is observed. In the nonlinear regime, a rotating dynamo calculation and a separate non-rotating simulation are found to saturate at a similar level, even though the mid-layer value of the local magnetic Reynolds number is smaller in the rotating case. We compute the Lyapunov exponents of the flow to show that the stretching properties of the convection are modified by rotation. Furthermore, rotation significantly reduces the magnetic energy dissipation in the lower part of the layer.

Self-similar turbulent boundary layer in pressure gradient. Four flow regimes

Self-similar flows in a turbulent boundary layer when the free-stream velocity is specified as a power, with the exponent m, function of longitudinal coordinate are investigated. The self-similar formulation not only simplifies solving of the problem by reducing the equations of motion to ordinary differential equations but also provides a mean for formulating closure conditions. It is shown that for the class of flows under consideration that depend on three governing parameters the dimensionless mixing length is a function of the normalized distance from the wall and the exponent m in the outer region and a universal function of local Reynolds number in the wall region, the latter corollary being true even when the skin friction vanishes. In calculations this function is set to be independent of pressure gradient, which gives the results very close to experimental data.There exist four different self-similar flow regimes. Each regime is related to its similarity parameter, one of which is the well-known Clauser equilibrium parameter and the other three are established for the first time. In case of adverse pressure gradient when the exponent m lies within certain limits, which depend on Reynolds number, the problem has two solutions with different values of the boundary layer thickness and the skin friction, which points out the possibility of hysteresis in near-separating flow. Separation occurs not at the minimal value of m that corresponds to the strongest adverse pressure gradient but at m = −0.216 − 0.4Rep−1/3 + O(Rep−2/3), where Rep is the Reynolds number based on the longitudinal pressure gradient. The results of the theory are in good agreement with experimental data.