Prandtl's Mixing Length Hypothesis Research Articles

A mixing length model for estimating channel conveyance

This paper describes a simple, physically based mixing length model that explains the functional form of Manning's equation for mean velocity in open channels. Manning's equation has been used to describe mean velocity for over 100 years and is essentially an empirical result rather than being based on an understanding of physical processes. The model described in this paper uses Prandtl's mixing length hypothesis, with mixing length modelled at each point within the cross-section being proportional to the distance to the nearest solid boundary. The model solves equations for the along-stream velocity field using a simple numerical method on regular and irregular finite-difference meshes. The results of the model are compared with Manning's equation and the Colebrook–White formula, giving good agreement across a range of channel sizes, roughnesses and geometries. The results and comparison are used to draw useful insights into open channel flows.

波面上の気流分布に対する対数則分布の適用性に関する考察

Air flow distributions and turbulent properties in the boundary layer over wind waves have been investigated by using PIV. Two regions adapted a logarithmic law were found in a vertical distribution of stream wise mean air flow. One of the logarithmic law regions is close to wavy surface and another called the middle layer is located upon the lower logarithmic law region. The friction velocity calculated from the middle layer corresponded to the Reynolds stress in a constant stress layer close to wavy surface region. The mixing length, l calculated from the experimental data showed that there were larger vortices than Prandtl's mixing length hypothesis, l = κy close to wavy surface for Case03 and 04. It was considered that the larger vortices were generated by air flow separation from wave crests.

A Mathematical Model of Turbulent Convective Fluid Flow Past a Vertical Infinite Plate with Hall Current

A mathematical model of magnetohydrodynamics (MHD) turbulent boundary layer fluid flow past a vertical infinite plate in a dissipative fluid with Hall current is considered. The plate is impulsively started and the flow problem is analysed thereafter. The effect of Hall current on the convectively cooled or convectively heated plate in the turbulent boundary layer is investigated. The Reynolds stresses, arising due to turbulence, in the momentum equations are resolved using Prandtl mixing length hypothesis. The governing equations for the problem are solved by a finite difference method. An analysis of the effects of Hall current, viscous dissipation and time on velocity and temperature profiles is done with the aid of graphs. It is found that primary velocity increases with increase in Hall parameter.

A rheological model for fluid flows with arbitrary Reynolds numbers

An approach to constructing a phenomenological model of motion for a viscous fluid alternative to the Prandtl mixing-length hypothesis is suggested. The approach makes it possible to describe the motion of a fluid independently of the regime realized in a given region of the flow. On the basis of this approach, a differential one-parameter model for the flow of a viscous fluid applicable to any regimes of motion, called a uniform laminar-turbulent model, is constructed. For this purpose, the field of a scalar turbulence measure is introduced, which equals the ratio of the Reynolds stress to the total stress in the case of a simple shear flow. This makes it possible to write new expressions for turbulent viscosity. The influence of the turbulence measure field on the flow is taken into account by using an additional transport differential equation. The model is applicable to both compressible and incompressible fluids and makes it possible to obtain solutions in quadratures for steady simple shear flows. Various forms of the system of equations of motion and boundary conditions are given.

Heat transfer and fluid flow during keyhole mode laser welding of tantalum, Ti–6Al–4V, 304L stainless steel and vanadium

Because of the complexity of several simultaneous physical processes, most heat transfer models of keyhole mode laser welding require some simplifications to make the calculations tractable. The simplifications often limit the applicability of each model to the specific materials systems for which the model is developed. In this work, a rigorous, yet computationally efficient, keyhole model is developed and tested on tantalum, Ti–6Al–4V, 304L stainless steel and vanadium. Unlike previous models, this one combines an existing model to calculate keyhole shape and size with numerical fluid flow and heat transfer calculations in the weld pool. The calculations of the keyhole profile involved a point-by-point heat balance at the keyhole walls considering multiple reflections of the laser beam in the vapour cavity. The equations of conservation of mass, momentum and energy are then solved in three dimensions assuming that the temperatures at the keyhole wall reach the boiling point of the different metals or alloys. A turbulence model based on Prandtl's mixing length hypothesis was used to estimate the effective viscosity and thermal conductivity in the liquid region. The calculated weld cross-sections agreed well with the experimental results for each metal and alloy system examined here. In each case, the weld pool geometry was affected by the thermal diffusivity, absorption coefficient, and the melting and boiling points, among the various physical properties of the alloy. The model was also used to better understand solidification phenomena and calculate the solidification parameters at the trailing edge of the weld pool. These calculations indicate that the solidification structure became less dendritic and coarser with decreasing weld velocities over the range of speeds investigated in this study. Overall, the keyhole weld model provides satisfactory simulations of the weld geometries and solidification sub-structures for diverse engineering metals and alloys.

A Model for Velocity Profile in Turbulent Boundary Layer with Drag Reducing Surfactants

Abstract A new model for mean velocity profile of turbulent water flow with added drag-reducing surfactants is presented in this paper. The general problem of drag due to frictional resistance is reviewed and drag reduction by the addition of polymers or surfactants is introduced. The model bases on modified Prandtl's mixing length hypothesis and includes three parameters, which depend on additives and can be evaluated by numerical simulation from experimental datasets. The advantage of the model in comparison with previously reported models is that it gives uniform curve for whole pipe section and can be determined for a new surfactant with less necessary measurements. The use of the model is demonstrated for surfactant Habon-G as an example.

Effects of Inlet Swirl Velocity on Static Characteristics of Annular Plain Seals. Numerical Analysis Based on Averaged Navier-Stokes Equations With Turbulent Coefficients.

Averaged Navier-Stokes equations with turbulent coefficients and the continuity equation are applied to a theoretical investigation of the effects of inlet swirl velocity on the static characteristics of annular plain seals employed in pumps. The turbulent coefficients are calculated based on the law of the wall and Prandtl's mixing length hypothesis. The Navier-Stokes equations and the continuity equation are numerically solved in consideration of the eccentricity of the rotor in the seal. The numerical results show that the inlet swirl velocity significantly influences the circumferential fluid velocity, which increases as the inlet swirl velocity increases in the direction of the rotor spin. For the eccentric seals, this phenomenon enhances the wedge action induced by the circumferential flow, and yields large hydrodynamic fluid-film pressure peaks and tangential fluid-film force. Hence, the effects of the inlet swirl velocity on the fluid-film pressure and the fluid-film force in the seal clearance qualitatively correspond to the effects of the rotor spinning velocity on these characteristics.

Effects of Inlet Swirl Velocity on Dynamic Characteristics of Annular Plain Seals. Numerical Analysis Based on Averaged Navier-Stokes Equations With Turbulent Coefficients.

Averaged Navier-Stokes equations with turbulent coefficients and the continuity equation are applied to a theoretical investigation of the effects of inlet swirl velocity on the dynamic characteristics of annular plain seals employed in pumps. The turbulent coefficients are calculated based on the "law of the wall" and Prandtl's mixing length hypothesis. The numerical results show that for any eccentricity ratio at an equilibrium position of the rotor center and for any rotor spinning velocity, the inlet swirl velocity significantly influences the cross-coupled stiffness terms, whose magnitudes increase as the inlet swirl velocity increases in the direction of the rotor spin. This phenomenon yields an increase of the whirl-frequency ratio, resulting in a narrower stable range of the rotor spinning velocity.

A generalisation of Prandtl's model for 3D open channel flows

The paper presents generalisation of the old Prandtl mixing-length hypothesis (MLH) for three-dimensional (3D) flows to provide engineers with a new tool to calculate the mean velocity distribution in turbulent open channel flows. These flows are usually not isotropic, i.e. turbulence may have different length scales in different directions. Hence, at every point of the flow a symmetrical second-order tensor with dimension of length should be given. The components of this tensor are specified based on the structure of the largest turbulent eddies. To demonstrate the usefulness of the proposed model, the Reynolds equations with the new turbulence model are solved and calculated streamwise velocity contours are compared with the measurement results. The importance of turbulent structure (sizes of the largest eddies and their growth rate) in forming the streamwise velocity contours is shown.

A generalisation of Prandtl's model for 3D open channel flows

The paper presents generalisation of the old Prandtl mixing-length hypothesis (MLH) for three-dimensional (3D) flows to provide engineers with a new tool to calculate the mean velocity distribution in turbulent open channel flows. These flows are usually not isotropic, i.e. turbulence may have different length scales in different directions. Hence, at every point of the flow a symmetrical second-order tensor with dimension of length should be given. The components of this tensor are specified based on the structure of the largest turbulent eddies. To demonstrate the usefulness of the proposed model, the Reynolds equations with the new turbulence model are solved and calculated streamwise velocity contours are compared with the measurement results. The importance of turbulent structure (sizes of the largest eddies and their growth rate) in forming the streamwise velocity contours is shown.

An optimised short bearing theory for nonlinear dynamic analysis of turbulent journal bearings

An approximate method for nonlinear dynamic analysis of turbulent journal bearings supporting an unbalanced rigid shaft is proposed. The method is based on two assumptions: the separation of variables for pressure and a parabolic pressure distribution in the axial direction of the bearing. To take into account inertia effects, the well-known algebraic turbulent model based on the Prandtl mixing length hypothesis is used. Using the Constantinescu's approach, the pressure equation is modified by introducing two turbulent coefficients which are depending on the local Reynolds number. The nonlinear equations of motion for the rotor-bearing system are solved by means of the Euler's scheme, and the journal centre trajectories are examined for cases with and without unbalance forces. To illustrate the validity of the present study, three cases of journal bearings are analysed. The accuracy of the minimum film thickness, the dynamic transmissibility coefficient and the peak-to-peak displacement amplitudes obtained by the proposed methodology are comparable to the more elaborate and time consuming 2-D finite difference solution, while the turbulent journal centre orbits are comparable to those obtained experimentally. It is concluded that the optimised short bearing theory shows the advantage of minimising the computation time required for nonlinear dynamic analysis of laminar and turbulent journal bearings without any significant loss of accuracy.

A higher-order asymptotic theory for fully developed turbulent flow in smooth pipes

A complete second-order asymptotic theory for fully developed turbulent flow in smooth pipes at high turbulent Reynolds numbers is presented in the paper. The theory is based on Prandtl's mixing-length hypothesis involving a fourth-order polynomial representation for the mixing length and taking into account its dependence on the Reynolds number. Two main contributions with respect to the existing literature have been achieved:

Discharge prediction in straight compound channels using the mixing length concept

A method for predicting the depth-discharge relationship in a compound channel is developed and applied to two different sets of experimental results. The method uses a mixing length formulation to account for the turbulent interaction between the main channel and the floodplain and the resulting momentum exchange. This momentum transfer tends to reduce the discharge in the main channel and increase the discharge on the floodplain. The net effect is a reduction in the overall discharge capacity of the compound channel. As a result, practical methods which can allow for the interaction effect are needed. In this formulation a variation of Prandtl's mixing length hypothesis is applied to calculate the apparent shear stresses, indicative of the turbulent interaction, on the sides of small vertical elements which comprise the compound channel cross-section. The approach suggested is to use the mixing length approximation to calculate the correction for the momentum interaction effects that are neglected when ...

Prediction of turbulent swirling flows in axisymmetric annuli

When a fluid system possesses a dominant flow direction, so that axial diffusion terms may be deleted from the governing equations on an order-of-magnitude basis, the flow may be simulated by an appropriate algorithm making a single or repeated sweeps of the domain marching in the flow direction. Provided that the upstream behavior does not influence the flow significantly, as in the case of weakly swirling and noncirculating flows, the Navier-Stokes equations can be approximated by a set of parabolic partial differential equations. These reduced equations may be efficiently and accurately solved by a stepwise integration in the predominant flow direction starting from prescribed downstream initial conditions. The use of parabolic equations, rather than the elliptic Navier-Stokes equations, has the advantage of decreasing the computer time and storage required. In this paper the single marching technique is employed to predict turbulent swirling flow along an annulus of constant cross-sectional area, using the constant eddy viscosity (CEV) and the Prandtl mixing length hypothesis (MLH) turbulence models. For low and medium swirl flows the computed results compared well with the available data.

LDV Investigations of Supersonic Turbulent Mixing Layer. (Single-Stream Seeding Experiments).

Turbulence characteristics of the supersonic turbulent mixing layer were investigated using a two-component LDV. The high-speed stream was of Mach number M1= 2.24 and low-speed one was of M2=1.61 (convective Mach number Mc=0.18, and free-stream velocity ratio r=0.86). To examine the behavior of fluid entrained into the mixing layer, single-stream seeding experiments were conducted ; either a high-speed stream or low-speed one was seeded. In the fully developed regions of the mixing layer, it was found that the mean streamwise velocity of fluid within the mixing layer, which originated from the high-speed stream, was not equal to that of the fluid from the low-speed stream. The transverse locations in which the turbulence intensity and Reynolds stress showed maximum values for the high-speed-stream seeding case was not coincident with that of the low-speed seeding case. They were also compared with that of the incompressible mixing layer case. The growth rate of the mixing layer was almost same as that of the incompressible case. The mixing length based on Prandtl's mixing length hypothesis was also discussed.

Investigations of mean and turbulent flow characteristics of a two dimensional plane diffuser

This paper reports the investigation of mean and turbulent flow characteristics of a two-dimensional plane diffuser. Both experimental and theoretical details are considered. The experimental investigation consists of the measurement of mean velocity profiles, wall static pressure and turbulence stresses. Theoretical study involves the prediction of downstream velocity profiles and the distribution of turbulence kinetic energy using a well tested finite difference procedure. Two models, viz., Prandtl's mixing length hypothesis and k-ɛ model of turbulence, have been used and compared. The nondimensional static pressure distribution, the longitudinal pressure gradient, the pressure recovery coefficient, percentage recovery of static pressure, the variation of Umax/Ubar along the length of the diffuser and the blockage factor have been valuated from the predicted results and compared with the experimental data. Further, the predicted and the measured value of kinetic energy of turbulence have also been compared. It is seen that for the prediction of mean flow characteristics and to evaluate the performance of the diffuser, a simple turbulence model like Prandtl's mixing length hypothesis is quite adequate.

Calculation of turbulent convection between corotating disks in axisymmetric enclosures

A numerical investigation is conducted for the case of turbulent flow in the unobstructed space between a pair of centrally clamped coaxial disks corotating in a fixed axisymmetric enclosure. The finite difference procedure of Chang et al. ( J. Heat Transfer 111, 625–632 (1989)) is extended to include a standard two-equation ( κ- ε) model of turbulence in the core of the flow. A van Driest relation, that accounts for the effects of streamline curvature and wall shear on the energy-containing length scales, is used in conjunction with Prandtl's mixing length hypothesis to model the near wall flow. The set of equations is solved assuming a constant property, circumferentially symmetric, statistically stationary flow. This approach predicts mean velocity and heat transfer results that are in good agreement with timeaveraged experimental data. The predictions reveal a flow that, in non-dimensional variables, tends to a limiting asymptotic state at high Reynolds numbers. In the absence of blowing a symmetrical pair of crossstream eddies appear near the enclosure wall the rotation of which increases with increasing disk rotational speed. High rotational speeds, in excess of 2400 rpm in the configuration studied, and small disk separations induce large values of shear and temperature (due to viscous dissipation) in the vicinity of the enclosure wall. The flow and its heat transfer characteristics can be drastically altered by the combined effects of radial and axial blowing. Specifically, it is shown that axial blowing significantly reduces the shear and heat transfer at the curved enclosure wall.

NUMERICAL SIMULATION OF LOCAL NUSSELT NUMBER FOR TURBULENT FLOW IN A SQUARE DUCT WITH A 180° BEND

Numerical predictions of the local Nusselt number for the curved three-dimensional incompressible turbulent flow of air with heat transfer in a square-sectioned duct with a 180° bend are presented and compared with experiment. The results are obtained using a finite-volume discretization of the Reynolds and energy equations employed in a semielliptic algorithm that marches repeatedly through the computational domain until convergence is reached. Results for each of three wall treatments that model the flow and heat trasnsfer in the thermally important near-wall region are compared. The first treatment uses Prandtl's mixing length hypothesis with van Driest's damping function in a fine near-wall grid approach. The other treatments are “two-layer” approaches in which near-wall effects are integrated for a relatively coarse near-wall mesh. The two two-layer models are constructed to allow diffusion of turbulent kinetic energy in near-wall regions and are therefore more general than “local-equilibrium” treatm...

Numerical modelling of turbulent flow through conical diffusers with uniform and wake velocity profiles at the inlet

This paper is mainly concerned with the modelling of turbulent flow through conical diffusers to determine the various boundary layer characteristics. Flow through 4.5° and 6.0° diffusers has been considered with uniform as well as wake velocity profiles at the inlet. A finite difference method has been used to solve the partial differential equation of u-momentum. For the solution, a marching integration technique with the tridiagonal matrix algorithm is used. The Prandtl mixing length hypothesis is used to model the turbulence. The velocity profiles, the displacement thickness, momentum thickness, shape factor, blockage factor, performance and effectiveness of these diffusers at various downstream stations have been predicted and compared with the available experimental results. The agreement in general is quite satisfactory.

Turbulent combustion of hydrogen in a boundary jet discharged into a co-current supersonic air flow

A numerical analysis was made of the ignition and combustion of low-temperature hydrogen in a turbulent boundary jet discharged through a plane slit into a co-current supersonic flow of heated air. The turbulent, chemically nonequilibrium flow is described by Reynolds' system of averaged equations of motion in a boundary-layer approximation generalized to the case of a reactive multicomponent mixture. The approach used to describe the properties of a turbulent flow is connected with the concept of additional eddy viscosity. Two models of turbulence were used: an algebraic model developed for high-speed turbulent boundary jets which is based on the Prandtl mixing-length hypothesis; and the differential two-parameter model of Jones and Launder. The results of numerical analysis of a boundary jet of hydrogen discharged at sonic velocity into a co-current supersonic flow were compared with data from direct measurements to evaluate the models.