Analysis of Inclined Wall Plume by the k-ε Turbulence Model

1990 ◽  
Vol 57 (2) ◽  
pp. 455-465 ◽  
Author(s):  
Yusuke Fukushima ◽  
Norio Hayakawa
Keyword(s):  
Similarity Solution ◽  
Flow Direction ◽  
Vertical Wall ◽  
Slope Angle ◽  
Similarity Solutions ◽  
Momentum Balance ◽  
Wide Range ◽  
Horizontal Wall ◽  
Wall Plume ◽  
The Mathematical Model

The similarity solution of inclined wall plume is obtained analytically. The mathematical model used herein consists of the continuity equation of flow, the momentum balance equation in the flow direction, the diffusion equation of concentration, the equation of kinetic energy of turbulence and the equation of viscous dissipation rate of turbulence. It is shown that this set of equations has the similarity solution which can be solved numerically for each angle of the inclined wall. This numerical model is applied to the wide range of the slope angle, which includes the vertical wall plume as the special case and the nearly horizontal wall plume. The velocity and concentration profiles of the inclined wall plume are explained well by the similarity solutions.

Influence of Upstream Conditions and Gravity on Highly Inertial Thin-Film Flow

2008 ◽  
Vol 130 (6) ◽  
Author(s):  
Roger E. Khayat
Keyword(s):  
Thin Film ◽  
Free Surface ◽  
Flow Field ◽  
Film Flow ◽  
Flow Direction ◽  
Vertical Wall ◽  
Galerkin Projection ◽  
Thin Film Flow ◽  
Thin Film Equations ◽  
Horizontal Wall

Steady two-dimensional thin-film flow of a Newtonian fluid is examined in this theoretical study. The influence of exit conditions and gravity is examined in detail. The considered flow is of moderately high inertia. The flow is dictated by the thin-film equations of boundary layer type, which are solved by expanding the flow field in orthonormal modes in the transverse direction and using Galerkin projection method, combined with integration along the flow direction. Three types of exit conditions are investigated, namely, parabolic, semiparabolic, and uniform flow. It is found that the type of exit conditions has a significant effect on the development of the free surface and flow field near the exit. While for the parabolic velocity profile at the exit, the free surface exhibits a local depression, for semiparabolic and uniform velocity profiles, the height of the film increases monotonically with streamwise position. In order to examine the influence of gravity, the flow is studied down a vertical wall as well as over a horizontal wall. The role of gravity is different for the two types of wall orientation. It is found that for the horizontal wall, a hydraulic-jump-like structure is formed and the flow further downstream exhibits a shock. The influence of exit conditions on shock formation is examined in detail.

Similarity solutions and viscous gravity current adjustment times

2019 ◽  
Vol 874 ◽  
pp. 285-298 ◽  
Author(s):  
Thomasina V. Ball ◽  
Herbert E. Huppert
Keyword(s):  
Differential Equation ◽  
Similarity Solution ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Gravity Current ◽  
Similarity Solutions ◽  
Partial Differential ◽  
Initial Shape ◽  
Wide Range

A wide range of initial-value problems in fluid mechanics in particular, and in the physical sciences in general, are described by nonlinear partial differential equations. Recourse must often be made to numerical solutions, but a powerful, well-established technique is to solve the problem in terms of similarity variables. A disadvantage of the similarity solution is that it is almost always independent of any specific initial conditions, with the solution to the full differential equation approaching the similarity solution for times $t\gg t_{\ast }$, for some $t_{\ast }$. But what is $t_{\ast }$? In this paper we consider the situation of viscous gravity currents and obtain useful formulae for the time of approach, $\unicode[STIX]{x1D70F}(p)$, for a number of different initial shapes, where $p$ is the percentage disagreement between the radius of the current as determined by the full numerical solution of the governing partial differential equation and the similarity solution normalised by the similarity solution. We show that for any initial shape of volume $V,\unicode[STIX]{x1D70F}\propto 1/(\unicode[STIX]{x1D6FD}V^{1/3}\unicode[STIX]{x1D6FE}_{0}^{8/3}p)$ (as $p\downarrow 0$), where $\unicode[STIX]{x1D6FD}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/(3\unicode[STIX]{x1D707})$, with $g$ representing the acceleration due to gravity, $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ the density difference between the gravity current and the ambient, $\unicode[STIX]{x1D707}$ the dynamic viscosity of the fluid that makes up the gravity current and $\unicode[STIX]{x1D6FE}_{0}$ the initial aspect ratio. This framework can used in many other situations, including where it is not an initial condition (in time) that is studied but one valid for specified values at a special spatial coordinate.

scholarly journals Advanced Constitutive Modeling of the Thixotropic Elasto-Visco-Plastic Behavior of Blood: Steady-State Blood Flow in Microtubes

Materials ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 367
Author(s):  
Konstantinos Giannokostas ◽  
Yannis Dimakopoulos ◽  
Andreas Anayiotos ◽  
John Tsamopoulos
Keyword(s):  
Blood Flow ◽  
Steady State ◽  
Constitutive Model ◽  
Blood Plasma ◽  
Constitutive Modeling ◽  
Flow Direction ◽  
Momentum Balance ◽  
Plastic Behavior ◽  
Flow Investigation ◽  
The Impact

The present work focuses on the in-silico investigation of the steady-state blood flow in straight microtubes, incorporating advanced constitutive modeling for human blood and blood plasma. The blood constitutive model accounts for the interplay between thixotropy and elasto-visco-plasticity via a scalar variable that describes the level of the local blood structure at any instance. The constitutive model is enhanced by the non-Newtonian modeling of the plasma phase, which features bulk viscoelasticity. Incorporating microcirculation phenomena such as the cell-free layer (CFL) formation or the Fåhraeus and the Fåhraeus-Lindqvist effects is an indispensable part of the blood flow investigation. The coupling between them and the momentum balance is achieved through correlations based on experimental observations. Notably, we propose a new simplified form for the dependence of the apparent viscosity on the hematocrit that predicts the CFL thickness correctly. Our investigation focuses on the impact of the microtube diameter and the pressure-gradient on velocity profiles, normal and shear viscoelastic stresses, and thixotropic properties. We demonstrate the microstructural configuration of blood in steady-state conditions, revealing that blood is highly aggregated in narrow tubes, promoting a flat velocity profile. Additionally, the proper accounting of the CFL thickness shows that for narrow microtubes, the reduction of discharged hematocrit is significant, which in some cases is up to 70%. At high pressure-gradients, the plasmatic proteins in both regions are extended in the flow direction, developing large axial normal stresses, which are more significant in the core region. We also provide normal stress predictions at both the blood/plasma interface (INS) and the tube wall (WNS), which are difficult to measure experimentally. Both decrease with the tube radius; however, they exhibit significant differences in magnitude and type of variation. INS varies linearly from 4.5 to 2 Pa, while WNS exhibits an exponential decrease taking values from 50 mPa to zero.

Robust sideslip angle observer of commercial vehicles based on cornering stiffness estimation using neural network

2022 ◽  
pp. 095440702110724
Author(s):  
Chenyu Zhou ◽  
Liangyao Yu ◽  
Yong Li ◽  
Jian Song
Keyword(s):  
Neural Network ◽  
Mathematical Model ◽  
Stability Control ◽  
Estimation Accuracy ◽  
Accurate Estimation ◽  
Commercial Vehicles ◽  
Sideslip Angle ◽  
The Neural Network ◽  
Wide Range ◽  
The Mathematical Model

Accurate estimation of sideslip angle is essential for vehicle stability control. For commercial vehicles, the estimation of sideslip angle is challenging due to severe load transfer and tire nonlinearity. This paper presents a robust sideslip angle observer of commercial vehicles based on identification of tire cornering stiffness. Since tire cornering stiffness of commercial vehicles is greatly affected by tire force and road adhesion coefficient, it cannot be treated as a constant. To estimate the cornering stiffness in real time, the neural network model constructed by Levenberg-Marquardt backpropagation (LMBP) algorithm is employed. LMBP is a fast convergent supervised learning algorithm, which combines the steepest descent method and gauss-newton method, and is widely used in system parameter estimation. LMBP does not rely on the mathematical model of the actual system when building the neural network. Therefore, when the mathematical model is difficult to establish, LMBP can play a very good role. Considering the complexity of tire modeling, this study adopted LMBP algorithm to estimate tire cornering stiffness, which have simplified the tire model and improved the estimation accuracy. Combined with neural network, A time-varying Kalman filter (TVKF) is designed to observe the sideslip angle of commercial vehicles. To validate the feasibility of the proposed estimation algorithm, multiple driving maneuvers under different road surface friction have been carried out. The test results show that the proposed method has better accuracy than the existing algorithm, and it’s robust over a wide range of driving conditions.

Countercurrent convection in a double-diffusive boundary layer

1985 ◽  
Vol 160 ◽  
pp. 181-210 ◽  
Author(s):  
R. H. Nilson
Keyword(s):  
Boundary Layer ◽  
Vertical Wall ◽  
Similarity Solutions ◽  
Relative Strength ◽  
Diffusive Boundary Layer ◽  
Buoyancy Forces ◽  
Diffusive Boundary ◽  
Laminar Convection ◽  
Similar Solutions ◽  
Double Diffusive

Countercurrent flow may be induced by opposing buoyancy forces associated with compositional gradients and thermal gradients within a fluid. The occurrence and structure of such flows is investigated by solving the double-diffusive boundary-layer equations for steady laminar convection along a vertical wall of finite height. Non-similar solutions are derived using the method of matched asymptotic expansions, under the restriction that the Lewis and Prandtl numbers are both large. Two sets of asymptotic solutions are constructed, assuming dominance of one or the other of the buoyancy forces. The two sets overlap in the central region of the parameter space; each set matches up with neighbouring unidirectional similarity solutions at the respective borderlines of incipient counterflow.Interaction between the buoyancy mechanisms is controlled by their relative strength R and their relative diffusivity Le. Flow in the outer thermal boundary layer deviates from single-diffusive thermal convection, depending upon the magnitude of the parameter RLe. Flow in the inner compositional boundary layer deviates from single-diffusive compositional convection, depending upon the magnitude of $RLe^{\frac{1}{3}}$.

scholarly journals Explicit Solutions of a Gravity-Induced Film Flow along a Convectively Heated Vertical Wall

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Ammarah Raees ◽  
Hang Xu
Keyword(s):  
Film Flow ◽  
Vertical Wall ◽  
Boussinesq Approximation ◽  
Convective Boundary Condition ◽  
Convective Boundary ◽  
Similarity Transformations ◽  
Reduced Equations ◽  
Homotopy Analysis ◽  
Gravity Driven ◽  
The Mathematical Model

The gravity-driven film flow has been analyzed along a vertical wall subjected to a convective boundary condition. The Boussinesq approximation is applied to simplify the buoyancy term, and similarity transformations are used on the mathematical model of the problem under consideration, to obtain a set of coupled ordinary differential equations. Then the reduced equations are solved explicitly by using homotopy analysis method (HAM). The resulting solutions are investigated for heat transfer effects on velocity and temperature profiles.

scholarly journals Stress anisotropy in natural debris flows during impacting a monitoring structure

Landslides ◽  
2021 ◽  
Author(s):  
Georg Nagl ◽  
Johannes Hübl ◽  
Roland Kaitna
Keyword(s):  
Debris Flows ◽  
Flow Direction ◽  
Vertical Wall ◽  
Force Plate ◽  
Frictional Resistance ◽  
Gravitational Mass ◽  
Analytical Models ◽  
Stress Anisotropy ◽  
Mass Flows ◽  
Stress Ratios

AbstractThe frictional resistance of rock and debris is supposed to induce stress anisotropy in the unsteady, non-uniform flow of gravitational mass flows, including debris flows. Though widely used in analytical models and numerical simulation tools, concurrent measurements of stresses in different directions are not yet available for natural flow events. The present study aims to investigate the relation of longitudinal and bed-normal stress exerted by two natural debris flows impacting a monitoring barrier in the Gadria creek, Italy. For that, a force plate in front of a barrier was used to continuously record forces normal to the channel bed, whereas load cells mounted on the vertical wall of the barrier recorded forces in flow direction. We observed an anisotropic stress state during most of the flow events, with stress ratios ranging between 0.1 and 3.5. Video recordings reveal complex deposition and re-mobilization patterns in front of the barrier during surges and highlight the unsteady nature of debris flows. These first-time in-situ measurements confirm the assumption of stress anisotropy in natural debris flows for gravitational mass flows, and provide data for model testing.

scholarly journals The Cardiac Electrophysiology Web Lab

2015 ◽  
Author(s):  
Jonathan Cooper ◽  
Martin Scharm ◽  
Gary R Mirams
Keyword(s):  
Cardiac Electrophysiology ◽  
Computational Modelling ◽  
Cell Types ◽  
Virtual Experiments ◽  
Online Resource ◽  
Voltage Curve ◽  
Current Voltage ◽  
Wide Range ◽  
Current Voltage Curve ◽  
The Mathematical Model

Computational modelling of cardiac cellular electrophysiology has a long history, with many models now available for different species, cell types, and experimental preparations. This success brings with it a challenge: how do we assess and compare the underlying hypotheses and emergent behaviours, in order to choose a model as a suitable basis for a new study, or characterize how a particular model behaves in different scenarios? We have created an online resource for the characterization and comparison of electrophysiological cell models under a wide range of experimental scenarios. The details of the mathematical model (quantitative assumptions and hypotheses formulated as ordinary differential equations) are separated from the experimental protocol being simulated. Each model and protocol is then encoded in computer-readable formats. A simulation tool runs virtual experiments on models, and a website – https://chaste.cs.ox.ac.uk/FunctionalCuration – provides a friendly interface, allowing users to store and compare results. The system currently contains a sample of 36 models and 23 protocols, including current-voltage curve generation, action potential properties under steady pacing at different rates, restitution properties, block of particular channels, and hypo-/hyper-kalaemia. This resource is publicly available, open source, and free; and we invite the community to use it and become involved in future developments. Those interested in comparing competing hypotheses using models can make a more informed decision; those developing new models can upload them for easy evaluation under the existing protocols, and even add their own protocols.

scholarly journals Natural convection air flow in vertical upright-angled triangular cavities under realistic thermal boundary conditions

2016 ◽  
Vol 20 (5) ◽  
pp. 1407-1420 ◽  
Author(s):  
Jaime Sieres ◽  
Antonio Campo ◽  
José Martínez-Súarez
Keyword(s):  
Natural Convection ◽  
Vertical Wall ◽  
Aperture Angle ◽  
Temperature Fields ◽  
Uniform Heat Flux ◽  
Thermal Boundary Conditions ◽  
Nusselt Numbers ◽  
Horizontal Wall ◽  
Rayleigh Numbers

This paper presents an analytical and numerical computation of laminar natural convection in a collection of vertical upright-angled triangular cavities filled with air. The vertical wall is heated with a uniform heat flux; the inclined wall is cooled with a uniform temperature; while the upper horizontal wall is assumed thermally insulated. The defining aperture angle ? is located at the lower vertex between the vertical and inclined walls. The finite element method is implemented to perform the computational analysis of the conservation equations for three aperture angles ? (= 15?, 30? and 45?) and height-based modified Rayleigh numbers ranging from a low Ra = 0 (pure conduction) to a high 109. Numerical results are reported for the velocity and temperature fields as well as the Nusselt numbers at the heated vertical wall. The numerical computations are also focused on the determination of the value of the maximum or critical temperature along the hot vertical wall and its dependence with the modified Rayleigh number and the aperture angle.

scholarly journals Similarity Solutions of the MHD Boundary Layer Flow Past a Constant Wedge within Porous Media

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Ramesh B. Kudenatti ◽  
Shreenivas R. Kirsur ◽  
Achala L. Nargund ◽  
N. M. Bujurke
Keyword(s):  
Boundary Layer ◽  
Porous Medium ◽  
Pressure Gradient ◽  
Adverse Pressure Gradient ◽  
Velocity Profiles ◽  
Similarity Solutions ◽  
Positive Pressure ◽  
Boundary Layer Equations ◽  
Similarity Transformations ◽  
Wide Range

The two-dimensional magnetohydrodynamic flow of a viscous fluid over a constant wedge immersed in a porous medium is studied. The flow is induced by suction/injection and also by the mainstream flow that is assumed to vary in a power-law manner with coordinate distance along the boundary. The governing nonlinear boundary layer equations have been transformed into a third-order nonlinear Falkner-Skan equation through similarity transformations. This equation has been solved analytically for a wide range of parameters involved in the study. Various results for the dimensionless velocity profiles and skin frictions are discussed for the pressure gradient parameter, Hartmann number, permeability parameter, and suction/injection. A far-field asymptotic solution is also obtained which has revealed oscillatory velocity profiles when the flow has an adverse pressure gradient. The results show that, for the positive pressure gradient and mass transfer parameters, the thickness of the boundary layer becomes thin and the flow is directed entirely towards the wedge surface whereas for negative values the solutions have very different characters. Also it is found that MHD effects on the boundary layer are exactly the same as the porous medium in which both reduce the boundary layer thickness.

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