Channel Of Arbitrary Cross Section Research Articles

Machine-learning-aided design optimization of internal flow channel cross-sections

• Artificial neural network-based models are developed for prediction of nusselt number and friction factor. • The predictions apply for fully developed laminar flow in constant cross-section ducts of different cross section shapes. • The machine learning (ML) model predictions are validated and compared against numerical simulations. • Use of ML models in shape optimization of channel cross section for different objective functions is demonstrated. • Guidelines for usage of the developed ML models are discussed. The design optimization of various thermal management components such as cold plates, heat sinks, and heat exchangers relies on accurate prediction of flow heat transfer and pressure drop. During the iterative design process, the heat transfer and pressure drop is typically either computed numerically or obtained using geometry-specific correlations for Nusselt number and friction factor. Numerical approaches are accurate for evaluation of a single design but become computationally expensive if many design iterations are required (such as during formal optimization processes). Correlation-based approaches restrict the design space to a specific set of geometries for which correlations are available. Surrogate models for the Nusselt number and friction factor, which are more universally applicable to all geometries than traditional correlations, would enable flexible and computationally inexpensive design optimization. The current work develops machine-learning-based surrogate models for predicting the Nusselt number and friction factor under fully developed internal flow in channels of arbitrary cross section and demonstrates use of these models for optimization of the cross-sectional channel shape. The predictive performance and generality of the machine learning surrogate models is first verified on various shapes outside the training dataset, and then the models are used in the design optimization of flow cross sections based on performance metrics that weigh both heat transfer and pressure drop. The optimization process leads to novel shapes outside the training data, and so numerical simulations are carried out on these optimized shapes to compare with the surrogate model predictions and show their performance is at least as good as that of shapes with known correlations available. A three-lobed shape was found to reduce friction factor, whereas a pentagon with rounded corners and an ice cream cone-shaped duct, both found using different performance metrics. Although the ML model predictions lose accuracy outside the training set for these novel shapes, the predictions follow the correct trends with parametric variations of the shape and therefore successfully direct the search toward optimized shapes.

Fast asymptotic-numerical method for coarse mesh particle simulation in channels of arbitrary cross section

Particles traveling through inertial microfluidic devices migrate to focusing streamlines. We present a numerical method that calculates migration velocities of particles in inertial microfluidic channels of arbitrary cross section by representing particles by singularities. Refinements to asymptotic analysis are given that improve the regularity of the singularity approximation, making finite element approximations of flow and pressure fields more effective. Sample results demonstrate that the method is computationally efficient and able to capture bifurcations in particle focusing positions due to changes in channel shape and Reynolds number.

Efficient modeling of three-dimensional convection–diffusion problems in stationary flows

A large number of transport processes in physics, chemistry, and engineering are described by a convection–diffusion equation. This equation is notoriously difficult to solve due to the presence of convection-related first-order gradient differential operators. We describe a new and efficient numerical method for solving the convection–diffusion equation for laminar flows within channels of arbitrary cross section. It is based on reducing the convection–diffusion equation to a set of pure diffusion equations with a complex-valued potential for which fast and numerically stable solvers are readily available. Additionally, we use an eigenvector projection method that allows us to compute snapshots of full concentration distributions over millions of finite elements within a few seconds using a conventional state-of-the-art desktop computer. Our results will be important for all applications where diffusion and convection are both important for correctly describing material transport.

Large Eddy Simulations of Turbulent Flows at Supercritical Pressure in a Vertical Heated Pipe Using Unstructured Cartesian Grids with Local Refinement

This paper presents the results of numerical simulation of turbulent flow of CO2 at supercritical pressure (SCP) in a uniform heated vertical tube using the large eddy simulation (LES) with the Smagorinsky subgrid scale (SGS) model. Unstructured Cartesian grids with local anisotropic refinement were used for modeling hydrodynamics and heat transfer in a circular pipe. Such grids unlike structured grids in cylindrical coordinates allow one to construct more optimal meshes for the LES method. In addition, such meshes allow one to simulate flows in channels of arbitrary cross-section. To verify the proposed methodology the results of the direct numerical simulation were used. Numerical simulations have been performed using the “in-house” CFD code ANES.

Large eddy simulations of air flow in a vertical heated pipe using unstructured Cartesian grids with local refinement

This paper presents the results of modeling of fully developed turbulent gas flow in a uniform heated vertical tube by using the LES method and Smagorinski model. Unstructured Cartesian grids with local anisotropic refinement were used for modeling hydrodynamics and heat transfer in cylindrical pipes. Such meshes unlike structured meshes in cylindrical coordinates allow to construct more detailed meshes for the LES method. In addition, such meshes allow to simulate flows in channels of arbitrary cross-section. To verify the proposed methodology the results of direct numerical simulation of turbulent air flow in a vertical heated tube for forced and mixed convection were used. Thus, for upward flows the most interesting case with significant laminarization flow due to the buoyancy influence was studied.

Rigorous theory for transient capillary imbibition in channels of arbitrary cross section

This article addresses a classical fluid mechanics problem where the effect of capillary action on a column of viscous liquid is analyzed by quantifying its time-dependent penetrated length in a narrow channel. Despite several past studies, a rigorous mathematical formulation of this inherently unsteady process is still unavailable, because these existing works resort to a crucial assumption only valid for mildly transient systems. The approximate theories use an integral approach where the penetration is described by equating total force acting on the domain to rate of change of total momentum. However, while doing so, the viscous resistance under temporally varying condition is assumed to be same as the resistance created by a quasi-steady velocity profile. Thus, leading order error appears due to such approximation which can only be true when the variation in time is not strong enough causing negligible transient deviation in the hydrodynamic quantities. The present paper proposes a new way to solve this problem by considering the unsteady field itself as an unknown variable. Accordingly, the analysis applies an eigenfunction expansion of the flow with unknown time-dependent amplitudes which along with the unsteady intrusion length are calculated from a system of ordinary differential equations. A comparative exploration identifies the situation for which the integral approach and the rigorous technique based on eigenfunction expansion deviate from each other. It also reveals that the two methods differ substantially in short-time dynamics at the initial stage. Then, an asymptotic perturbation shows how the two sets of results should coincide in their long-time behavior. In this way, the findings will provide a comprehensive understanding of the physics behind the transport phenomenon.

High-order accurate conservative method for computing the poiseuille rarefied gas flow in a channel of arbitrary cross section

A high-order accurate method is proposed for analyzing the isothermal rarefied gas flow in an infinitely long channel with an arbitrarily shaped cross section (Poiseuille flow). The basic idea behind the method is the use of hybrid unstructured meshes in physical space and the application of a conservative technique for computing the gas velocity. Examples of calculations are provided for channels of various cross sections in a wide range of Knudsen numbers. Schemes of the first-, second-, and third orders of accuracy in space are compared.

Pressure Drop in Rectangular Microchannels as Compared With Theory Based on Arbitrary Cross Section

Pressure driven liquid flow through rectangular cross-section microchannels is investigated experimentally. Polydimethylsiloxane microchannels are fabricated using soft lithography. Pressure drop data are used to characterize the friction factor over a range of aspect ratios from 0.13 to 0.76 and Reynolds number from 1 to 35 with distilled water as working fluid. Results are compared with the general model developed to predict the fully developed pressure drop in arbitrary cross-section microchannels. Using available theories, effects of different losses, such as developing region, minor flow contraction and expansion, and streaming potential on the measured pressure drop, are investigated. Experimental results compare well with the theory based on the presure drop in channels of arbitrary cross section.

The Poiseuille Flow in a Pipe with Temperature Dependent Viscosity and Prescribed Flow Rate

This note studies the stationary Poiseuille flow in a cylindrical channel of arbitrary cross-section with temperature dependent viscosity and internal dissipation. We assume the flow-rate given and we treat the axial pressure gradient as an unknown. Theorems of existence and uniqueness are proved using a special transformation.

Integration of the boussinesq equations for steady channel and jet flows

The equations of long nonlinear waves in round jets and channels of arbitrary cross section are considered with account for the transverse acceleration of the fluid particles (Boussinesq approximation). In the general case of steady flows, the equations in the form of shallow water equations with the pressure expressed in terms of the variational derivative of the kinetic energy of a thin transverse fluid layer, have three first integrals with three arbitrary constants. Examples of solutions of the equations for solitary capillary-gravitation waves in rectangular and triangular channels are presented and compared with the higher approximations. The shape of the free boundary of the round jet is determined. In the case of outflow from a conical nozzle an analytical dependence of the jet contraction ratio on the conicity angle is obtained. The dependence is in agreement with the experimental data for angles of less than 45°.

Hydraulic jump in circular and U-shaped channels

The Reynolds equations of mean turbulent flow in a two-dimensional open channel of arbitrary cross-section have been analyzed. An integral equation for the turbulent hydraulic jump is proposed. In the closure model adopted, the depth-averaged effective normal Reynolds stress is taken proportional to the product of constant eddy viscosity and depth-averaged axial velocity gradient, and the constant of proportionality that is independent of channel geometry. The general theory has been applied to the flows in circular and U-shaped channels. The solutions for sequent depth, hydraulic jump length, roller length and aeration length have been estimated. The comparison of the theory with experimental data of Hager and Stahl for circular and U-shaped channels give very encouraging results.

Structure of the turbulent hydraulic jump in a trapezoidal channel

The axial flow structure of turbulent hydraulic jump has been analysed and the general equation valid for a channel of arbitrary cross section has been proposed. Based on the Reynolds equations of mean turbulent motion in two dimensional steady incompressible flow subjected to hydrostatic pressure distribution, the integral equations of depth averaged flow over a channel of arbitrary cross sectional area are obtained. An integral method has been developed where inertia, pressure gradient and depth averaged normal Reynolds stress play the dominant role. The closure model for variation of depth averaged normal Reynolds stress has been expressed as product of the constant eddy viscosity and the gradient of the depth averaged axial velocity with respect to axial distance. In the trapezoidal channel the closed form solution for the upper surface profile and axial length of the hydraulic jump have been obtained. The comparison of the theory with experimental data is remarkably good. The theory shows that for F, larger than a fixed value, the surface profile approaches a limiting universal solution provided the variables are appropriately non-dimensionalized. Further, the present predictions on the roller length are also supported by experimental data in rectangular and triangular channels.

Asymptotic form of the admixture transport equation for a channel flow with an anisotropic diffusion tensor

The problem of the asymptotically correct reduction of a 3-D mass (heat) transfer equation to a 1-D equation in a flow with anisotropic diffusion properties is considered. The convective mass (heat) transfer domain is a cylindrical channel of arbitrary cross section. The diffusion coefficient matrix is assumed to be independent of the spatial coordinates. In the equivalent diffusion equation constructed, a certain effective diffusion (dispersion [1]) coefficient is introduced. Formulas for this coefficient are obtained. A relation between the effective diffusion coefficient calculations and the problem of minimization of a certain functional is established, i. e. the possibility of calculations based on variational methods is noted. An example of an exact calculation of the effective diffusion coefficient is considered. The possibility of a generalization of the problem, in which the effective diffusion (heat conduction) equation is essentially a nonlinear equation of general form for the one-dimensional case, is indicated.

Resonance phenomena for water waves in channels of arbitrary cross-section

This article studies the evolutionary problem for linear gravity waves on the surface of water in a uniform, symmetric channel which is excited by an antisymmetric pressure force of frequency ω at the free surface. It is shown that there is a countably infinite set of frequencies {ω0, ω1, …} which give rise to resonance phenomena: the amplitude of the wave motion grows like t1/2 as t→∞ in a sense which is precisely specified. Under pressure forcing at any other frequency the solution obeys the principle of limiting amplitude. These results are obtained by combining methods developed for problems in acoustic waveguides with regularity theory for elliptic boundary-value problems in non-smooth domains. Copyright © 1999 John Wiley & Sons, Ltd.

Trapped-mode solutions for gravity-capillary water waves in channels of arbitrary cross-section

This article establishes the existence of a trapped-mode solution to a linearized water-wave problem. The fluid occupies a symmetric horizontal channel that is uniform everywhere apart from a confined region which either contains a thin vertical plate spanning the depth of the channel or has indentations in the channel walls; the forces of gravity and surface tension are operative. A trapped mode corresponds to an eigenvalue of the composition of an inverse differential operator and a Neumann-Dirichlet operator for an elliptic boundary-value problem in the fluid domain. The existence of such an eigenvalue is established by extending previous results dealing with the case when surface tension is absent.

On the Existence of Trapped Modes in Channels of Arbitrary Cross-section

This article establishes the existence of trapped-mode solutions of a linearized water-wave problem. The fluid occupies a symmetric horizontal channel that is uniform everywhere apart from a confined region which either contains a thin vertical plate spanning the depth of the channel or has indentations in the channel walls. A trapped mode corresponds to an eigenvalue of a non-local Neumann-Dirichlet operator for an elliptic boundary-value problem in the fluid domain. The existence of such an eigenvalue is established by generalizing previous results concerning spectral theory for differential operators to this non-local operator.

Concerning the description of long nonlinear waves in channels

A method of deriving the equations that describe long nonlinear waves in channels of arbitrary cross section, taking the transverse acceleration of fluid particles into account (the Boussinesq approximation), is proposed. For channels of certain cross sections the equations are written in explicit form. In the case of narrow channels the Boussinesq equations and those of the next approximation are written in explicit form for arbitrary cross sections.

Model of Dam‐Break Floods for Channels of Arbitrary Cross Section

To obtain the relationship between the hydraulic characteristics and the breach shape, a new method is developed for calculation of the hydraulic characteristics of dam‐break waves in an arbitrary channel, whose perimeter can be described by a number of line segments. By a mathematical transformation, the shape parameters of a breach cross section are separated from the integral equation and become a coefficient that can be calculated. The original integral equation of dam‐break flood is transformed into a model equation. This model equation can be treated as a pure mathematical problem. By this model, the hydraulic characteristics of the dam‐break waves with complicated outlet shape can be calculated directly and the accuracy of the method is the same as that of Ritter's solution of a rectangular cross section. In addition to the solving process for an arbitrary section, an example of a complex cross section and the data for a trapezoidal cross section are presented.

Locally forced critical surface waves in channels of arbitrary cross section

Critical long surface waves forced by locally distributed external pressure applied on the free surface in channels of arbitrary cross section are studied in this paper. The fluid under consideration is inviscid and has constant density. The upstream flow is uniform and the upstream velocity is assumed to be near critical, i.e.,u 0=u c +eλ+0(e2), where 0<ɛ≪1 andu c is the critical velocity determined by the geometry of the channel. The external pressure applied on the free surface as the forcing isɛ 2 Pδ(x). Then the first order perturbation of the free surface elevation satisfies a forced Korteweg-de Vries equation (fK-dV). It is shown in this paper that: (i) If $$\lambda \geqslant \lambda _c = (3b^2 P^2 \alpha ^2 /( - 4\beta m_1^2 ))^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} > 0$$ (supercritical), the stationaryfK-dV has two cusped solitary wave solutions; (ii) if $$\lambda< \lambda _L = (3b^2 P^2 \alpha _2 /(\beta m_1^2 ))^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}< 0$$ (subcritical), the stationaryfK-dV has a downstream cnoidal wave solution; (iii) whenλ=λ L , the unique stationary solution of thefK-dV is a wave free hydraulic fall; (iv) ifλ=λ d =−λ L , thefK-dV has a jump solution; and (v) ifλ L <λ<λ c , thefK-dV does not have stationary solutions. Some free surface profiles and bifurcation diagrams are presented.

Lower estimates of the characteristic frequencies of the oscillations of a liquid with a free surface in channels of arbitrary cross-section

Lower estimates are obtained for the leading characeristic frequency of oscillations of a liquid in a channel of arbitrary cross-section with several sections of the free surface of the liquid. The case of oscillations in the plane of the cross-section of the channel is considered. The domain occupied by the cross-section can be multiply connected and bounded by a piecewise smooth curve. The derivation of the estimates is not connected with the need to find standard domains and is not based on variational methods /1–3/.