Prandtl Mixing Length Theory Research Articles

An approach on turbulent flow of pseudo-plastic nanofluids and heat transfer subject to wall slip

The purpose of this paper is to study the turbulent flow of pseudo-plastic nanofluids (PPNF) and heat transfer subject to wall slip. Four types of nanoparticles (Fe3O4, CuO, Cu, Ag) are taken into account in carboxymethyl cellulose (CMC)-water-based fluid. The Prandtl mixing length theory is introduced to divide the turbulent boundary layer into two regions: laminar sub-layer and turbulent region. The numerical solutions are obtained by bvp4c technique. Results show that the wall-slip parameter has important influence on velocity and temperature fields, the variations are more intensive in laminar sub-layer. The local friction coefficient decreases for larger wall-slip parameter and smaller fraction of nanoparticles. And the heat transfer is significantly strengthened (in Nusselt number) for smaller wall-slip parameter and larger fraction of nanoparticles. Moreover, for four different nanofluids, the Ag/CMC nanofluid is more conducive to enhance the turbulent heat transfer than other nanofluids.

Spatial wave solutions for generalized atmospheric Ekman equations

In this paper, we use Prandtl mixing-length theory and semiempirical theory to extend the classical problem of the wind in the steady atmospheric Ekman layer with constant eddy viscosity. New generalized atmospheric Ekman equations are established and qualitative properties of the corresponding ODEs are studied. Spatial wave solutions results for the nonlinear and implicit equations with different nonlinearities are presented.

Turbulent boundary layer heat transfer of CuO–water nanofluids on a continuously moving plate subject to convective boundary

Abstract The turbulent boundary layer (TBL) heat transfer of CuO–water nanofluids on a continuously moving plate subject to convective boundary are investigated. Five different shapes of nanoparticles are taken into account. Prandtl mixing length theory is adopted to divide the TBL into two parts, laminar sub-layer and turbulent region. The numerical solutions are obtained by bvp4c and accuracy is verified with previous results. It is found that the transfer of momentum and heat in the TBL is more obvious in laminar sub-layer than in turbulent region. The rise of velocity ratio parameter increases the velocity and temperature while decreases the local friction coefficient. The heat transfer increases significantly with the increase of velocity ratio parameter, Biot number, and nanoparticles volume fraction. For nanoparticles of different shapes, the heat transfer characteristics are Nu x (sphere) < Nu x (hexahedron) < Nu x (tetrahedron) < Nu x (column) < Nu x (lamina).

On fluid flow and heat transfer of turbulent boundary layer of pseudoplastic fluids on a semi-infinite plate

The boundary layer of a pseudoplastic fluid on a semi-infinite plate for a high generalized Reynolds number is analyzed. Based on the Prandtl mixing length theory, the turbulent region is divided into two regions. The coupled momentum and temperature equations, with a generalized thermal conductivity model, have made the process of finding the analytical solutions much difficult. By using the similarity transformation, the equations are converted to four ordinary differential equations constrained by ten boundary conditions. An interesting technique of scaling and translation of the calculation domain of one region into another is used to make the system of equations easier to solve. It is found that the fluid with a smaller power-law index, associated with a thinner velocity boundary layer thickness, processes a lower friction coefficient. Furthermore, the increase in the Reynolds number causes a thinner velocity boundary layer and a decreasing friction coefficient on the wall. Changes in temperature occur more slowly near the plate surface with a rise in the power-law index and a decrease in the Reynolds number.

Modeling of droplet dispersion in a turbulent Taylor–Couette flow

A process of dispersion of droplets in a developed turbulent Taylor-Couette flow is modeled. An apparatus, where the inner cylinder rotates whereas outer one is immobile, is considered. To exclude the effect of Taylor vorticities on a turbulent flow pattern, only flow regimes characterized by relatively high Reynolds numbers (>13,000) are investigated. Relatively dilute dispersions are modeled: the maximum droplet volume fraction does not exceed 10%. The time-dependent droplet size distribution in a Taylor-Couette device is described by an advection-diffusion equation containing a population balance term. Both breakup and coalescence of droplets are taken into account. An experimentally verified engineering model of a turbulent Taylor-Couette flow, based on the Prandtl mixing length theory, is used for calculating velocity, eddy diffusivity and turbulence energy dissipation rate distributions across a Taylor-Couette device gap. The known breakup and coalescence models of Coulaloglou and Tavlarides are employed for population balance modeling. However, the breakup model has been modified to account more accurately for turbulence – droplet interactions.The governing model equations are solved numerically. The computed droplet size distributions are compared with those obtained in the laboratory Taylor-Couette device in the presence of emulsion stabilizing surfactants for different rotation speeds and stirring durations. The dispersion process in a Taylor-Couette flow in both the presence and absence of surfactants is illustrated by numerical examples. The results of simulations and experiments are critically analyzed.

The effect of density gradient on boundary flow

Based on Prandtl's mixing-length theory, this study proposes new theoretical formulae of velocity profiles resulting from the forcing by a constant horizontal buoyancy gradient in unstratified and stably stratified flows. Based on the one-dimensional water column momentum equation, the vertical turbulent shearing stress profile is found to deviate from a linear distribution and follow a parabolic distribution, differing from that in neutral flow. The shearing stress curves upward with the current following the density gradient, and curves downward with the currents opposite to the density gradient. For a constant eddy viscosity, the well-known estuarine circulation is obtained through the parabolic shearing distribution. For a vertically parabolic eddy viscosity, the new-proposed velocity profile by Burchard & Hetland (2010) is obtained. In this paper, we estimate the viscosity profile based on Prandtl's mixing-length theory and then derive the new formulae of the velocity profiles. Through comparison with the numerical turbulence model, the velocity profiles presented in the paper agree well with the numerical results. In addition, the new velocity profiles can be applied to determine the valid range and evaluate errors of the log-fit in baroclinic boundary flows.

Research on the energy dissipation mechanism of nonobstructive particle dampers based on the dense granular flow theory

To study the energy dissipation mechanism of nonobstructive particle dampers (NOPDs) and provide guidance to the application of NOPDs, the dense granular flow theory was introduced to establish a quantitative energy dissipation model for NOPDs. The convection movement of the particles under vibrational excitations was studied using the discrete element method, and the Prandtl mixing length theory was adopted to modify the constitution law of dense granular flows. The pressure of the granular flow was obtained by equivalenting the vibrational excitation to a body force acted on particles. Theoretical results showed that the energy dissipation rate of the NOPD was increased with the vibration intensity and decreased with the granular diameter. It also indicated that particles near the side wall and the bottom of the damper dissipated more energy than those particles in other regions. The theoretical model was verified by simulation and experimental result. The results may provide a new approach to studying the energy dissipation mechanism of NOPD and give some guidance to enhancing the damping performance of NOPD in engineering practices.

Study on mass transfer of turbulent air flow sweeping liquid surface

Different forces in interface of gas–liquid in the process of turbulent air flow sweeping liquid surface were analyzed. Characteristics of flowing section were done and Prandtl mixing length theory was used, the principle of turbulent flow mass transfer in the process of turbulent air flow sweeping liquid surface was studied. Effect of Reynolds number, gas flow structure and Prandtl mixing length on the mass transfer were analyzed. It indicated that the turbulent flow mass diffusion coefficient and mass transfer coefficient were involved with the state of gas flow and was affected by gas flow structure. The results are in favor of lucubrate study and engineering application of mass transfer in gas–liquid interface.

Analytical estimation of mixing coefficient induced by surface wave-generated turbulence based on the equilibrium solution of the second-order turbulence closure model

Based on the equations of motion and the assumption that ocean turbulence is of isotropy or quasi-isotropy, we derived the closure equations of the second-order moments and the variation equations for characteristic quantities, which describe the mechanisms of advection transport and shear instability by the sum of wave-like and eddy-like motions and circulation. Given that ocean turbulence generated by wave breaking is dominant at the ocean surface, we presented the boundary conditions of the turbulence kinetic energy and its dissipation rate, which are determined by energy loss from wave breaking and entrainment depth respectively. According to the equilibrium solution of the variation equations and available data of the dissipation rate, we obtained an analytical estimation of the characteristic quantities of surface-wave-generated turbulence in the upper ocean and its related mixing coefficient. The derived kinetic dissipation rate was validated by field measurements qualitatively and quantitatively, and the mixing coefficient had fairly good consistency with previous results based on the Prandtl mixing length theory.

DERIVING THE SEMI-EMPIRICAL FORMULA TO COMPUTE THE FRICTION FACTOR λ FOR TURBULENT FLOW IN PIPE

Based on law of shear stress in turbulent flow. Prandd's mixing length theory, and Bakhmeteflfs point of view on "wall velocity", turbulent velocity distribution u on wetted area can be derived for smooth pipe and complete turbulence, rough pipe. Discharge Q and average velocity y are obtained, after the integration, Q= ∫∫wu.dw is done. Relying on the properties of uniform flow, relationship between V, friction factor λ, and shear velocity u is set up. After eliminating u*. velocity V is obtained as a function of Reynolds number Re or relative roughness e/D. Finally, the value of friction factor z can be derived as a function of Re or e/D for the two above-mentioned cases. These formations of z formulas are almost same as the experimental ones introduced by Nikuradse with minor deviations in the factors and their relative errors do not exceed 1% for smooth pipe, and 2% for complete turbulence, rough pipe. Through this research result, the rightness of Prandtl's mixing length theory is almost asserted.

Hydraulic jumps in a channel

We present a study of hydraulic jumps with flow predominantly in one direction, created either by confining the flow to a narrow channel with parallel walls or by providing an inflow in the form of a narrow sheet. In the channel flow, we find a linear height profile upstream of the jump as expected for a supercritical one-dimensional boundary layer flow, but we find that the surface slope is up to an order of magnitude larger than expected and independent of flow rate. We explain this as an effect of turbulent fluctuations creating an enhanced eddy viscosity, and we model the results in terms of Prandtl's mixing-length theory with a mixing length that is proportional to the height of the fluid layer. Using averaged boundary-layer equations, taking into account the friction with the channel walls and the eddy viscosity, the flow both upstream and downstream of the jump can be understood. For the downstream subcritical flow, we assume that the critical height is attained close to the channel outlet. We use mass and momentum conservation to determine the position of the jump and obtain an estimate which is in rough agreement with our experiment. We show that the averaging method with a varying velocity profile allows for computation of the flow-structure through the jump and predicts a separation vortex behind the jump, something which is not clearly seen experimentally, probably owing to turbulence. In the sheet flow, we find that the jump has the shape of a rhombus with sharply defined oblique shocks. The experiment shows that the variation of the opening angle of the rhombus with flow rate is determined by the condition that the normal velocity at the jump is constant.

Improvement of Prandtl mixing length theory and application in modeling of turbulent flow in circular tubes

In order to correctly predict tube cross section time-smoothed velocity distribution, friction factor and mass transfer behavior, two models for turbulent flow in circular tubes based on classical Prandtl mixing length theory and a modified mixing length were established. The results show that the modified mixing length includes the introduction of a damping function for the viscous sublayer and the second-order derivative to approximate eddy velocity. The calculated dimensionless time-smoothed velocity from the model based on Prandtl mixing length is much better than the result from the concept of eddy viscosity. The calculated eddy viscosity from the model based on modified mixing length is much better than the result from the model based on the classical Prandtl mixing length theory. And the friction factor calculated from the model based on the modified mixing length agrees well with the reported empirical relationships.

Modeling of turbulent heat transfer during the solidification process of continuous castings

Prandtl mixing length theory with average heat capacity was successfully implemented to model turbulent continuous casting (CC) process. This method features the simplicity of zero equation model along with ease of maintaining higher time step increment using average heat capacity method. A series of simulations were investigated to study the effect of inlet speed ( Pe), superheat ( Θ o), mold cooling rate ( Bi m) and post mold cooling rate ( Bi p) on the process characteristics and were presented in terms of viscosity contours, location and slope of the solidification front, axial velocity profiles, velocity vectors, local heat flux and shell thickness. The ranges of the parameters studied in the current work were: Pe = 3.5–6.1, Θ o = 1.2–2.0, Bi m = 0.2–0.4 and Bi p = 0.1–0.4. The effect of mold length ( L m) was also investigated to establish remedies of breakout due to higher Pe and Θ o. The current studies reveal that Pe, Θ o and L m have significant effect on the performance and productivity of the cast material.

A turbulent melt-lubrication model of surface wear in railgun armatures

This paper describes a melt-lubrication model of the liquid film interface in solid-armature railguns that includes the effects of turbulence. The liquid film is modeled as a high-speed Couette flow with viscous heating. The model focuses strictly on mechanical wear and does not include magnetohydrodynamic body forces or Joule heating. The effects of turbulence are incorporated by using Prandtl's mixing-length theory and semiempirical methods. Turbulent viscosity is added to the laminar viscosity for the solution of the momentum equations. Thermal resistance is accounted for at the thermal contact between the melt film and rail. Expressions are obtained for the quantities of film thickness and melt wear rate. Results from the model are compared with experimental data that measured high-speed mechanical wear of 7075 aluminum sliding against ETP copper for face pressures ranging from 45 to 150 MPa in railguns. Although the model reproduces general trends in the data, it predicts melt speeds that are significantly greater than measured; in this regard, a previously published laminar model provides a better fit. However, given the shortcomings of both models, it is quite possible that a purely thermal hydraulic formulation is inadequate to the task of modeling high-speed, high-pressure mechanical wear, and that viscoplastic processes in the slider should be considered next.

NUMERICAL INVESTIGATION OF THE EFFECTS OF TURBULENCE FROM SUBMERGED ENTRY NOZZLE DURING CONTINUOUS CASTING PROCESS

A numerical investigation was performed using Prandtl mixing-length theory along with the average heat capacity method to model a turbulent plane submerged entry nozzle continuous casting process. This method features the simplicity of a zero-equation model along with ease of maintaining the time-step increment. Results from the study match well with other published data. The current study indicates that submergence depth of the entry nozzle affects the turbulence level and hence controls the quality of the finished product. The inlet speed of the molten metal also controls the solidification front slope and the surface heat flux of the caster.

Basic Pattern in Atmospheric Turbulence Model**The project supported by National Natural Science Foundation of China under Grant Nos. 40175016 and 40305006

From the controlling equations of atmosphere motion, Prandtl's mixing length theory is used to derive the atmospheric turbulence models, such as Burgers equation model and Burgers-KdV equation model. And then the projective Riccati equations are applied to solve these atmospheric turbulence models, where much more patterns are obtained, including solitary wave pattern, singular pattern, and so on.

Wall shear stress measurement in a turbulent pipe flow using ultrasound Doppler velocimetry

A turbulent boundary layer of a water flow is investigated by means of pulsed ultrasound Doppler velocimetry. The advantage of this method is the acquisition of complete velocity profiles along the sound propagation line within very short time intervals. The shear stress velocity, used for normalizing the velocity profiles, was determined by fitting the profiles to the universal profiles in a turbulent boundary layer obtained from Prandtl's mixing length theory. A coordinate transformation in the near-wall region is proposed to allocate the velocity data to "true" wall distances. From the experimental values of the wall shear stress velocity, the friction factors for a turbulent pipe flow are calculated and compared to the Blasius law. The overall error in measurement was estimated to ±8.4%.

Multicomponent-Fuel Film-Vaporization Model for Multidimensional Computations

A multicomponent-fuel e lm-vaporization model is developed to be used in multidimensional spray and combustion computations. For the gas phase the vaporization rate was evaluated using the turbulent boundary-layer assumption and the Prandtl mixing-length theory. A third-order polynomial was used to model the temperature and species concentration proe les within the liquid e lm in order to predict accurate surface temperature and surface mass fractions, which are crucial to evaluating the species vaporization rates. By this approach the governing equations for the e lm were reduced to a set of ordinary differential equations. The new model offers a signie cant reduction in computational cost and sufe cient accuracy compared to solving the governing equations for the e lm directly. The new model was verie ed against exact numerical solutions with excellent agreement for several cases concerning the vaporization process of a e lm on a e at plate. The results were also compared with the solutions obtained using an ine nite-diffusion model. The new model predicted the vaporization history more accurately than the ine nite-diffusion model. Finally, the new model was applied to study the e lm evolution for a spray/wall impingement case, and physical insight was gained from the study.

The Influence of Negative Viscosity on Wind-Driven, Barotropic Ocean Circulation in a Subtropical Basin

Abstract In this paper, a new barotropic wind-driven circulation model is proposed to explain the enhanced transport of the western boundary current and the establishment of the recirculation gyre. In this model the harmonic Laplacian viscosity including the second-order term (the negative viscosity term) in the Prandtl mixing length theory is regarded as a tentative subgrid-scale parameterization of the boundary layer near the wall. First, for the linear Munk model with weak negative viscosity its analytical solution can be obtained with the help of a perturbation expansion method. It can be shown that the negative viscosity can result in the intensification of the western boundary current. Second, the fully nonlinear model is solved numerically in some parameter space. It is found that for the finite Reynolds numbers the negative viscosity can strengthen both the western boundary current and the recirculation gyre in the northwest corner of the basin. The drastic increase of the mean and eddy kinetic en...

Pressure Drop and Heat Transfer in Turbulent Duct Flow: A Two-Parameter Variational Method

A two-parameter variational method is introduced to calculate pressure drop and heat transfer for turbulent flow in ducts. The variational method leads to a Galerkin-type solution for the momentum and energy equations. The method uses the Prandtl mixing length theory to describe turbulent shear stress. The Van Driest model is compared with experimental data and incorporated in the numerical calculations. The computed velocity profiles, pressure drop, and heat transfer coefficient are compared with the experimental data of various investigators for fully developed turbulent flow in parallel plate ducts and pipes. This analysis leads to development of a Green’s function useful for solving a variety of conjugate heat transfer problems.