Solution Of Laminar Flow Research Articles

Laminar flow past an enclosed and open backward-facing step

Numerical solutions for laminar flow past an enclosed and open backward-facing step are presented. The semi-implicit method for pressure equations (SIMPLE) methodology is used with finite volume discretization. The investigation concentrated on the differences in the lower reattachment length which are caused by the upper boundary layer. The upper boundary layer can cause three major effects; flow retardation, inhibition of perpendicular streamlines, and an upper recirculation region. These all affect the lower reattachment length.

Heat Transfer and Pressure Drop Correlations for Twisted-Tape Inserts in Isothermal Tubes: Part I—Laminar Flows

Laminar flow correlations for f and Num are developed based on experimental data for water and ethylene glycol, with tape inserts of three different twist ratios. The uniform wall temperature condition is considered, which typifies practical heat exchangers in the chemical and process industry. These and other available data are analyzed to devise flow regime maps that characterize twisted-tape effects in terms of the dominant enhancement mechanisms. Depending upon flow rates and tape geometry, the enhancement in heat transfer is due to the tube partitioning and flow blockage, longer flow path, and secondary fluid circulation; fin effects are found to be negligible in snug- to loose-fitting tapes. The onset of swirl flow and its intensity is determined by a swirl parameter, Sw=Resw/y, that defines the interaction between viscous, convective inertia, and centrifugal forces. Buoyancy-driven free convection that comes into play at low flow rates with large y and ΔTw is shown to scale as Gr/Sw2≫ 1. These parameters, along with numerical baseline solutions for laminar flows with y = ∞, are incorporated into correlations for f and Num by matching the appropriate asymptotic behavior. The correlations describe the experimental data within ±10 to 15 percent, and their generalized applicability is verified by the comparison of predictions with previously published data.

Performance of externally pressurized conical thrust bearing under laminar and turbulent flow conditions

In the present theoretical investigation, the effect of turbulent lubrication, represented by Reynolds stresses, on the performance of externally pressurized circular and conical thrust bearings is studied. A recent algebraic Reynolds stress model is employed. The governing equations are solved iteratively using the finite difference method. Both the inertia forces and thermal effects are neglected in the present analysis. The effects of recess radius ratio, film thickness ratio and Poiseuille and Couette Reynolds numbers on the performance characteristics of the circular and conical bearings are studied. The bearings' performances under laminar flow condition are also presented and compared with the turbulent condition. The present study reveals that the circular bearing has slightly higher dimensionless pressure, load and torque than the conical bearing. Further, the turbulent flow solution gives higher dimensionless pressure, load and torque than the laminar flow solution. Finally, increasing the film thickness ratio or the recess radius ratio has the effect of decreasing both the dimensionless load and torque.

Similarity solutions for laminar thin-film flow along a vertical wall

The laminar accelerating flow of a thin film along a vertical wall is examined by using similarity principles and an empirical relation for the boundary layer thickness in terms of the entrance length parameter and the interaction parameter is suggested, which will be useful for comparison of results obtained by any other simple approximate solutions.

Commercial turbofan engine exhaust nozzle flow analyses

The recently developed three-dimensional code is able to perform a computational investigation of complex aircraft aerodynamic components. This code was developed for solving the simplified Reynolds-averaged NavierStokes equations in a three-dimensional multiblock/mul tizone structured mesh domain. The present analysis was applied to commercial turbofan exhaust flow systems. Solution sensitivity to grid density is presented. Laminar flow solutions were developed for all grids, and two-equation k-£ solutions were developed for selected grids. Static pressure distributions, mass flow, and thrust quantities were calculated for on-design engine operating conditions. Good agreement between predicted surface static pressures and experimental data was observed at different locations. Mass flow was predicted within 0.2% of experimental data. Thrust forces were typically within 0.6% of experimental data.

3-D implicit Navier-Stokes solver for internal turbulent compressible flows

A new code aimed at the solution of three-dimensional internal laminar and turbulent compressible flows is presented. The solution procedure is based on the scalar form of the approximate factorization method. The Navier-Stokes equations in mass-averaged form are iterated in time until a steady solution is reached. Evidence is given to the implicit treatment of diffusive terms that proved able to speed up convergence and enhance the algorithm robustness. Turbulence effects are accounted for by the Baldwin-Lomax turbulence model and the q - ω two-equation model. For the first, an investigation on the model behavior in case of multiple boundaries is performed. The numerical code is validated by computing the flow in the Stanitz 90-deg turning elbow. Finally the code is applied to the computation of the flow in the VKI LS-82 three-dimensional linear cascade.

Two-dimensional Navier-Stokes equations with adaptivity on structured meshes

Adaptivity techniques employed on two-dimensional structured multiblock meshes are examined. They include mesh point enrichment, mesh movement and their combination. In all approaches, flow variables are taken for the adaptivity criteria. Steady state solutions for inviscid and viscous laminar flows are obtained by the finite volume method, based on the Jameson Runge-Kutta scheme. Some applications for turbulent flows are also provided. All three adaptivity techniques significantly improve the accuracy with which flow features, such as shock waves and viscous boundary layers are captured. The employment of the same integration method allows for a reliable comparison between the proposed techniques. Although the study has been performed on structured meshes, the application to unstructured and hybrid meshes is straightforward.

The numerical solution of laminar flow in a re-entrant tube geometry by a Chebyshev spectral element collocation method

A Chebyshev spectral element collocation method is used for the numerical solution of laminar flow in a re-entrant tube geometry. The high accuracy of the method allows the study of the merging process of the vortices in the re-entrant region. Results are presented for values of the Reynolds number up to 400 and for different lengths of the re-entrant tube lip. For low Reynolds number, the observed behaviour of the flow in the re-entrant region may be interpreted by analytical results on Stokes flow from the literature. As the Reynolds number is increased, a vortex that grows in size with the Reynolds number, is detected below the tip of the re-entrant lip.

Numerical dissipation in F3D thin-layer Navier-Stokes solution for flows with wall transpiration

Numerical boundary conditions for mass injection/suction at the wall are incorporated in the thin-layer Navier-Stokes code, F3D. The accuracy of the boundary conditions and the code is assessed by a detailed comparison of the predictions of velocity distributions and skin-friction coefficients with exact similarity solutions for laminar flow over a flat plate with variable blowing/suction, and measurements for turbulent flow past a flat plate with uniform blowing. In laminar flow, F3D predictions for friction coefficient compare well with exact similarity solution with and without suction, but the code produces large errors at moderate-to-large values of blowing. These errors are attributed to the numerical dissipation due to Steger-Warmin g flux-vector splitting in the upwinding (streamwise) direction. This numerical dissipation is shown to yield a slight Mach number dependence of skin-friction coefficient due to blowing in turbulent flow. Predicted surface pressures for turbulent flow past an airfoil with mass injection are in qualitative agreement with measurements for a flat plate.

Explicit Runge-Kutta method for three-dimensional internal incompressible flows

A computer code has been developed to obtain the steady-state solutions for three-dimensional laminar incompressible flow governed by the Navier-Stokes equations. A central difference finite-volume formulation with an explicit one-step multistage Runge-Kutta time-stepping scheme for the primitive variables (i.e., velocity components and pressure) is used. The preconditioning matrix method, which is a generalized version of the artificial compressibility method, is used to construct a pressure equation from the continuity equation. Fourth-order artificial dissipation is used to suppress high-frequency oscillations

A Method of “Separation of Variables” for the Solution of Laminar Boundary-Layer Equations of Narrow-Channel Flows

A method of solution for laminar channel flow is established using the von Mises transformation followed by the “separation” of the main flow velocity into amplitude and profile functions. The known boundary conditions on the latter enable a generalized solution that yields, through a parametric relation between auxiliary parameters (the characteristics), a system of ordinary differential equations for the velocity amplitude and the pressure. The result is a powerful semi-analytical method which is very easy to implement for a variety of internal flow configurations. The method is, in essence, a downstream general solution which may be extended upstream to the singular limit of a uniform main flow velocity distribution at the channel entrance. The regular solution for uniform initial velocity distribution cannot thus be obtained. Comparisons with other solutions show only qualitative agreement, for reasons which are discussed, whereas agreement with experimental results, made in a separate publication, is remarkably good.

Unsteady hypersonic boundary layers for slender axisymmetric bodies with large injection rates

A semi-similar solution of an unsteady hypersonic laminar compressible boundary layer flow over a slender axisymmetric body with massive blowing has been obtained when the free stream velocity varies arbitrarily with time. The governing partial differential equations have been solved numerically by combining the implicit finite difference scheme with the quasi-linearization technique. The results have been obtained for (i) an accelerating/decelerating stream and (ii) a fluctuating stream. The skin friction responds to the fluctuation in the free stream more compared to the heat transfer. It is observed that the effect of large injection (blowing) rates is to move the viscous boundary layer away from the surface. The effect of the variation of the density-viscosity product across the boundary layer is found to be negligible for large blowing rates. Massive blowing reduces significantly the values of skin friction and heat transfer but the effect of the transverse curvature parameter is just reverse. Location of the dividing streamline increases as injection rate increases, but decreases with the increase of the transverse curvature parameter.

Pertubation solution of non-newtonian fluid axial laminar flow through eccentric annuli

Using the perturbation method, the axial laminar flow of Non-Newtonian fluid through an eccentric annulus is studied in the present paper. The relative eccentricity e is taken as a perturbation parameter, and the first order perturbation solutions of the problem, such as velocity field, limit velocity and pressure gradient, are all obtained.

Incompressible steady aerodynamics using a standard finite element code

Data were also acquired at a - 4-deg angle of attack, which allowed more complete measurement of pressure side heat transfer values. Comparing the -4-deg (pressure side) data with the + 4-deg (suction side) data showed that for the roughened airfoil the pressure side experienced lower heat transfer than the suction side. In computer codes, heat transfer from airfoils is often estimated by using cylinder-in-crossflow heat transfer values in the leading edge region and flat plate heat transfer values farther aft. Figure 6 shows the IRT data for the dense 2, 0-deg, without spray condition compared with the cylinder and flat plate heat transfer values. The heat transfer in the stagnation region for the dense 2 roughened airfoil agrees fairly well with Frossling's5 smooth cylinder laminar flow solution. Moving downstream on the airfoil, the heat transfer drastically increases, reaching a maximum level near s/c of 0.035, and then decreases to a level fairly consistent with turbulent flow flat plate heat transfer values.6 The measured Frossling numbers at specific Reynolds numbers are somewhat higher than their respective flat pjate turbulent values. However, the higher measured heat transfer may be due to the increase in surface area caused by the roughness elements (3-7% increase on each gauge for the dense roughness patterns) that was not taken into account in the data analysis. It may be mentioned here that the maximum heat transfer is in the same general region, if slightly aft, of ice horn growth observed during glaze ice accretion.7 Conclusions Local heat transfer measurements from a roughened NACA 0012 airfoil were successfully obtained in flight and in the NASA Lewis icing research tunnel using the method and apparatus described in this work. Major conclusions resulting from this study are as follows. 1) The addition of roughness to the airfoil surface drastically increased the heat transfer downstream of stagnation. The roughness elements disturbed the laminar boundary-layer flow and in some cases caused a transition to turbulent flow. 2) Comparison of the flight and tunnel roughened surface data showed that the general effect of increased turbulence was a slight increase in heat transfer, especially at the higher Reynolds numbers. 3) Generally, the roughened surface airfoil cases showed the suction side heat transfer monotonically increasing with angle of attack.

Efficient solution of two-dimensional steady separated flows

This work is concerned with the numerical solution of two-dimensional incompressible steady laminar separated flows at moderate-to-high values of Re. The vorticity-stream function Navier-Stokes equations, as well as approximate models based upon the boundary-layer theory, will be considered. The main objective of the paper is to present the development of an efficient approach for solving a class of problems usually referred to as “high Re weakly separated flows”. A description is given of a block-alternating-direction-implicit method, which applies the approximate factorization scheme of Beam and Warming to the vorticity-stream function equations, using the delta form of the deferred correction procedure of Khosla and Rubin to combine the stability of upwind schemes with the accuracy of central differences. The logical steps which led to the development of a more efficient incremental block-line Gauss-Seidel method and to a simple multigrid strategy particularly suited for this kind of numerical scheme are then outlined. Finally, benchmark-quality solutions for separated flows inside diffusers and channels with smooth as well as sudden expansions are presented.

Conforming Chebyshev Spectral Collocation Methods for the Solution of Laminar flow in a Constricted Channel

The numerical simulation of steady planar two-dimensional, laminar flow of an incompressible fluid through an abruptly contracting channel using spectral domain decomposition methods is described. The key features of the method are the decomposition of the flow region into a number of rectangular subregions and spectral approximations which are pointwise C(1) continuous across subregion interfaces. Spectral approximations to the solution are obtained for Reynolds numbers in the range 0 to 500. The size of the salient corner vortex decreases as the Reynolds number increases from 0 to around 45. As the Reynolds number is increased further the vortex grows slowly. A vortex is detected downstream of the contraction at a Reynolds number of around 175 that continues to grow as the Reynolds number is increased further.

Numerical solution of laminar flow generated in an annulus by rotating screens

The governing equation and the appropriate boundary condition describing stationary laminar flow in a curved channel and in an annulus with one (upper) and two (upper and lower) rotating screens, were solved numerically by finite-difference method. In the curved channel multiple solutions were obtained in accordance with the predictions of previous theoretical and experimental investigations. In contrast to that, no multiple solutions were found for the flow in an annulus, neither with two nor with one rotating screen. The numerically computed axial velocity distributions in annulus were compared to the corresponding experimental profiles measured in a turbulent flow of a homogeneous fluid created in annulus by one or two rotating screens. The qualitative agreement between the results was unexpectedly good.

Chebyshev solution of laminar boundary layer flow

An expansion procedure using the Chebyshev polynomials is proposed by using El-Gendi method [1], which yields more accurate results than those computed by P. M. Beckett [2] and A. R. Wadia and F. R. Payne [6] as indicated from solving the Falkner-Skan equation, which uses a boundary value technique. This method is accomplished by starting with Chebyshev approximation for the highest-order derivative and generating approximations to the lower-order derivatives through integration of the highest-order derivative.

Dispersion of Solute in Laminar Flow through a Circular Tube

Abstract An exact closed-form solution of a mathematical model describing the transport of soluble matter in a solvent inside a circular tube has been obtained in terms of a confluent hypergeometric function and shown to be in excellent agreement with previously published experimental and numerical works.

Numerical solution for laminar two-dimensional flow about a fixed and transversely oscillating cylinder in a uniform stream

A finite element solution is presented for laminar 2-dimensional flow past a fixed and transversely oscillating cylinder. The solution of vortex street development behind the cylinder is obtained when the cylinder remains fixed. Then the cylinder which is assumed elastically mounted is allowed to oscillate in the cross-flow direction under the action of the fluctuating lift force. The grid system is translated with the cylinder at each time step and the field variables are interpolated to new grid locations. The computer results predict the lock-in phenomenon which occurs when oscillation frequency is near the natural vortex-shedding frequency. Particular attention was given to flow configuration, by means of streamlines, filament lines, and equi-vorticity lines. The effect of cylinder oscillation on lift and drag forces and separation angles is extracted from the numerical results. Computer results at Reynolds numbers 106 and 115 are in good agreement with experimental measurements conducted by the author.