Brinkman Forchheimer Equations Research Articles

Moderate deviation principle for the 2D stochastic convective Brinkman–Forchheimer equations

ABSTRACT This work is concerned about some asymptotic behaviour of the two-dimensional stochastic convective Brinkman–Forchheimer (SCBF) equations, which describes the motion of incompressible viscous fluid through a rigid, homogeneous, isotropic, porous medium. We establish a central limit theorem and a moderate deviation principle for the SCBF equations using a variational method (based on weak convergence approach) developed by Budhiraja and Dupuis.

A Fully-Mixed Formulation for the Steady Double-Diffusive Convection System Based upon Brinkman–Forchheimer Equations

We propose and analyze a new mixed finite element method for the problem of steady double-diffusive convection in a fluid-saturated porous medium. More precisely, the model is described by the coupling of the Brinkman–Forchheimer and double-diffusion equations, in which the originally sought variables are the velocity and pressure of the fluid, and the temperature and concentration of a solute. Our approach is based on the introduction of the further unknowns given by the fluid pseudostress tensor, and the pseudoheat and pseudodiffusive vectors, thus yielding a fully-mixed formulation. Furthermore, since the nonlinear term in the Brinkman–Forchheimer equation requires the velocity to live in a smaller space than usual, we partially augment the variational formulation with suitable Galerkin type terms, which forces both the temperature and concentration scalar fields to live in $$\mathrm {L}^4$$ . As a consequence, the aforementioned pseudoheat and pseudodiffusive vectors live in a suitable $$\mathrm {H}(\mathrm {div})$$ -type Banach space. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram and Banach–Necas–Babuska theorems, allow to prove the unique solvability of the continuous problem. As for the associated Galerkin scheme we utilize Raviart–Thomas spaces of order $$k\ge 0$$ for approximating the pseudostress tensor, as well as the pseudoheat and pseudodiffusive vectors, whereas continuous piecewise polynomials of degree $$\le k + 1$$ are employed for the velocity, and piecewise polynomials of degree $$\le k$$ for the temperature and concentration fields. In turn, the existence and uniqueness of the discrete solution is established similarly to its continuous counterpart, applying in this case the Brouwer and Banach fixed-point theorems, respectively. Finally, we derive optimal a priori error estimates and provide several numerical results confirming the theoretical rates of convergence and illustrating the performance and flexibility of the method.

A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations

AbstractWe propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree $k$ for the pseudostress tensor and discontinuous piecewise polynomial elements of degree $k$ for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters.

Numerical analysis on laminar forced convection improvement of hybrid nanofluid within a U-bend pipe in porous media

Nanofluids and porous media have offered a feasible approach to develop heat transfer within industrial systems. The application of porous substances with high thermal conductivity can enhance the heat transfer as well. Besides the heat transfer improvements, porous media are capable of enhancing the pressure drop. In this regard, the current research is aimed at conducting a 3D numerical investigation on the laminar flow and heat transfer of Al2O3- CuO- water hybrid nanofluid inside a U-bend pipe within a porous medium. The evaluations were performed for a wide range of governing parameters. ANSYS-FLUENT is a finite volume-based computational fluid dynamics (CFD) solver which was considered to discretize the governing equations. The line-by-line method was also applied for iterative solution of the algebraic equations. As well as, the nanofluid flow was employed as a two-phase flow where a Darcy–Brinkman–Forchheimer equation was exploited to model fluid flow within a porous medium. Simulations were carried out under the laminar flow regime using a finite volume scheme. Moreover, the thermal boundary conditions of the cylinder walls were constant uniform heat flux. The averaged Nusselt number, volume fraction (0%–5%), and the performance evaluation criteria (PEC) were assessed for a verity of Darcy and Dean numbers. The findings of the current work were compared with the experimental data and the simulation accuracy was verified due to the proper agreement between the theoretical and experimental results. Based on the results, Darcy number (10−4 –10−1) and the porous thickness ratio enlargement could significantly affect the elevation of the heat transfer coefficient. Furthermore, for all nanofluids, the averaged Nusselt number and pressure drop exhibited an increasing pattern upon an increment of volume fraction while PEC showed a decreasing one. In addition, the maximum PEC was observed in configurations encompassing a permeable porous media (i.e. a medium with Da = 0.1 and rP = 0.8).

Design and optimization of a high-temperature latent heat storage unit

High-temperature latent heat thermal energy storage systems offer compact storage solutions, benefiting from the large latent heat of phase change materials. Common challenges in their design process are the low thermal conductivity of the phase change material and the low convective heat transfer between the phase change material and the working fluid. We present a design methodology for a high-temperature latent heat thermal energy storage unit. Eutectic Si-Mg with high thermal conductivity is considered as the phase change material, which is encapsulated in vertical SiC tubes. Charging/discharging is achieved by passing a heat transfer fluid along the encapsulation. In some cases, the flow passage is filled with porous medium to enhance the convective heat transfer between the encapsulated phase change material and the heat transfer fluid. A transient multi-physics model was developed to analyze the performance of the storage unit. The model couples the Brinkman-Forchheimer equations for fluid flow and a local thermal non-equilibrium formulation for the heat transfer, including a P1 approximation to consider the radiation in the porous domain. The apparent heat capacity formulation was used for the domain containing phase change material. A parametric study was conducted to evaluate the unit performance’s sensitivity. These results were used in a multi-criteria optimization to guide the design and sizing of tailored storage units. The results show that practical high-temperature latent heat storage units with an effectiveness as large as 0.95 and maximum storage energy of about 810 MJ/m3 can be achieved. The investigated tubular configuration can be scaled into a thermal energy storage system through in-series or parallel addition of multiple units, providing a straightforward approach to a tailored storage solution.

Magnetic field effects on forced convection flow of a hybrid nanofluid in a cylinder filled with porous media: a numerical study

The magnetic field can serve as a proper controlling parameter for heat transfer and fluid flow; it can be also employed to maximize the thermodynamic efficiency in various fields. Nanofluids and porous inserts are among the conventional approaches of heat transfer enhancements. Porous media, in addition to improving the heat transfer, can enhance the pressure drop. This research presents a numerical investigation on the magnetohydrodynamics forced convection effects of Al2O3–CuO–water nanofluid inside a partitioned cylinder within a porous medium. The calculations were carried out for a broad range of governing parameters. The nanofluid flow is modeled as a two-phase flow using two-phase mixture model, and the Darcy–Brinkman–Forchheimer equation is employed to model fluid flow in porous media. Simulation was also conducted under the laminar flow regime by finite volume method. Furthermore, the thermal boundary condition of constant uniform heat flux was imposed on the cylinder walls. The average Nusselt number as well as the performance evaluation criteria (PEC) were examined for diverse Darcy numbers (0.0001 < Da < 0.1) and Hartmann numbers (0 < Ha < 40). The results indicate the significant impact of Hartmann and Darcy number enhancement on the elevation of heat transfer coefficient. Additionally, incorporation of nanoparticles to the base fluid increased the PEC in all cases. Moreover, the PEC reached to its maximum value in configurations involving permeable porous media (i.e., a medium with Da = 0.1 and Ha = 40).

Natural convection of Nanoliquid from a Cylinder in Square Porous Enclosure using Buongiorno\u2019s Two-phase Model

Natural convection of nanoliquid in a square porous enclosure has been studied using non homogeneous two-phase Buongiorno’s model. The outer of enclosure has cold temperature and a circular cylinder is put at the center. A finite heated segment is located on the top cylinder surface which is otherwise insulated. The momentum in the porous layer is modeled applying the Brinkman-Forchheimer equations. The analysis are conducted in the following interval of the associated groups: the portion of heated surface (5% ≤ H ≤ 100%), the concentration (0.0 ≤ ϕ ≤ 0.04), the Darcy number, 10−5 ≤ Da ≤ 10−2 and the cylinder size, (0.15 ≤ R ≤ 0.25). The minimum heat transfer rate of the active surface were obtained at location ξ = 90°. In general, the ratio of the heat transfer per unit area of the heat source decreases as the length of the heated surface increases. The heat transfer rate is intensified for the half thermally active surface and high value of Darcy number at higher nanoparticles concentration.

The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay

&lt;p style='text-indent:20px;'&gt;This paper concerns the stability of pullback attractors for 3D Brinkman-Forchheimer equation with delays. By some regular estimates and the variable index to deal with the delay term, we get the sufficient conditions for asymptotic stability of trajectories inside the pullback attractors for a fluid flow model in porous medium by generalized Grashof numbers.&lt;/p&gt;

Non-autonomous 3D Brinkman-Forchheimer equation with singularly oscillating external force and its uniform attractor

For $\rho\in [0, 1)$ and $\varepsilon \gt 0$, the non-autonomous 3D Brinkman-Forchheimer equation with singularly oscillating external force $ \begin{align*} &u_t-\nu\Delta u+au+b|u|u+c|u|^\beta u+\nabla p = f_0(x,t)+\varepsilon^{-\rho}f_1(x,\frac{t}{\varepsilon}),\\ &\mathrm{div}u = 0 \end{align*} $ are considered, together with the averaged equation $ \begin{align*} &u_t-\nu\Delta u+au+b|u|u+c|u|^\beta u+\nabla p = f_0(x,t),\\ &\mathrm{div}u = 0 \end{align*} $ formally corresponding to the limiting case $\varepsilon = 0$. First, within the restriction $\rho \lt 1$ and under suitable translation-compactness assumptions on the external forces, the uniform (w.r.t.$\varepsilon$) boundedness of the related uniform attractors $\mathcal{A}^\varepsilon$ is established when $1 \lt \beta\leq 4/3$. This fact is not at all intuitive, since in principle the blow up of the oscillation amplitude might overcome the averaging effect due to the term $\frac{t}{\varepsilon}$ in $f_1$. Next, the convergence of the attractor $\mathcal{A}^\varepsilon$ of the first equation to the attractor $\mathcal{A}^0$ of the second one as $\varepsilon\rightarrow 0^+$ is established.

Forced convection in a fluid saturated anisotropic porous channel with isoflux boundaries

Fully developed forced convective flow inside a channel filled with a porous material bounded by two impermeable walls subject to a constant heat flux is considered. We consider the Brinkman-Forchheimer equation to govern the flow inside the porous medium, which accounts for the presence of the inertial term. We assume that the porous medium is anisotropic in nature and the permeability is varying along all the directions so that it appears as a positive semidefinite matrix in the momentum equation. We have obtained velocity, temperature, and Nusselt number numerically due to the presence of the nonlinear quadratic term in the momentum equation. Asymptotic solutions for small Darcy number (∼10−3) and high Darcy number (∼10) are obtained. The asymptotic behavior of the Nusselt number is discussed. The key purpose of this paper is to study the effect of anisotropic permeability ratio, anisotropic angle, and inertial parameter on the hydrodynamic quantities and heat transfer for the configuration considered. In particular, we observe that for the moderate range of Darcy number, 10−2 to 102, inertia plays a significant role in the Nusselt number. We observe that inclusion of anisotropic permeability enhances the relative heat transfer rate by almost 20% compared to the corresponding isotropic situation. We present a detailed analysis about the inclusion of the permeability matrix in the Brinkman-Forchheimer extended Darcy momentum equation.

Optimization of an additively manufactured functionally graded lattice structure with liquid cooling considering structural performances

Effective cooling or heat exchange is a typical engineering issue related with various industrial products, and the lattice structure fabricated by additive manufacturing is expected to be useful for effective liquid cooling. Moreover, such a heat exchanger demands structural performances such as stiffness and small thermal deformation when, for example, casting die and transporters of heated objects. Thus, in this research, we develop an optimization method for lattice volume fraction distribution using lattice structure approximation and a gradient method considering three coupled physical problems: fluid flow, thermal conduction, and convection and linear elasticity. Fluid flow is approximated by deriving effective properties from the Darcy–Forchheimer law and analyzing the flow according to the Brinkman–Forchheimer equation. The basis lattice shape is formed as three orthogonally connected pillars. The effective performance of a representative dimension unit lattice was calculated based on the statically averaging theorem and the relationship between the design variable and effective properties were approximated by polynomial functions. Two types of optimization problems were considered: maximization of fluid cooling performance under strain energy constraint and unconstrained minimization of normal direction of the loading and heating surface. The validity of the proposed methodology was investigated through three-dimensional examples. Although observable errors in accuracy exist between results obtained from optimization and full-scale models, the relative performance optimization was considered successful.

Dynamics and stability of the 3D Brinkman–Forchheimer equation with variable delay (I)

Dynamics and stability of the 3D Brinkman–Forchheimer equation with variable delay (I)

Method to optimize an additively-manufactured functionally-graded lattice structure for effective liquid cooling

The development of cooling devices is important for many industrial products, and the lattice structure fabricated by additive manufacturing is expected to be useful for effective liquid cooling. However, lattice density should be carefully designed for an effective arrangement of coolant flow. In this research, we optimize the lattice density distribution using a lattice structure approximation and the gradient method. Fluid flow is approximated by deriving effective properties from the Darcy–Forchheimer law and analyzing the flow according to the Brinkman–Forchheimer equation. Thermal conduction and convection are also approximated as a weakly coupled problem. We use a simple basic lattice shape composed of pillars, optimizing only its density distribution by setting the pillar diameter as the design variable. Steady-state pressure and temperature reductions are treated as multi-objective functions. Through 2D and 3D numerical studies, we discuss the validity and limitations of the proposed method. Although observable errors in accuracy exist between the results obtained from the optimization and full scale models, relative performance optimization was considered successful.

Mixed finite element for two-dimensional incompressible convective Brinkman-Forchheimer equations

In this work, the two-dimensional convective Brinkman-Forchheimer equations are considered. The well-posedness for the variational problem and its mixed finite element approximation is established, and the error estimates based on the conforming approximation are obtained. For the computation, a one-step Newton (or semi-Newton) iteration algorithm initialized using a fixed-point iteration is proposed. Finally, numerical experiments using a Taylor-Hood mixed element built on a structured or unstructured triangular mesh are implemented. The numerical results obtained using the algorithm are compared with the analytic data, and are shown to be in very good agreement. Moreover, the lid-driven problem at Reynolds numbers of 100 and 400 is considered and analyzed.

Solution of Darcy–Brinkman–Forchheimer Equation for Irregular Flow Channel by Finite Elements Approach

The finite element method of solution with curved triangles to solve the three-dimensional, fully-developed Darcy–Brinkman–Forchheimer flow equation in channel with curved side is solved using quasi-linearization and Gauss-Seidel iteration method. Exhaustive numerical computation and numerical experimentation reveals the parameters’ influence on the velocity distributions. A salient feature of the method adopted in the present paper is that it ensures that the errors are almost equally distributed among all the nodes. It is found that the irregular cross-section channel with upward concave boundary decelerates the flow. Numerical experimentation involved different order curved triangular elements and extensive computation revealed that the quintic order curved triangular element yields the desired solution to an accuracy of 10−5. The finite element method is found to be very effective in capturing boundary and inertia effects in the three-dimensional, fully-developed flow through porous media. Further, it prevails with regards to giving the required answer for vast estimations of Forchheimer number when shooting technique fails to do as such. The technique can be effortlessly utilized in some other sporadic cross-area channel.

Enhanced electroosmotic flow and ion selectivity in a channel patterned with periodically arranged polyelectrolyte-filled grooves

An enhanced electroosmotic flow through a surface-modulated microchannel is considered. The microchannel is modulated by periodically arranged rectangular grooves filled with polyelectrolyte materials. The flat surface of the walls is maintained at a constant charge density. A nonlinear model based on the Poisson–Nernst–Planck equations coupled with the Darcy–Brinkman–Forchheimer equation in the polyelectrolyte region and Navier–Stokes equations in the clear fluid region is adopted. Going beyond the widely employed Debye–Huckel linearization, we adopt numerical methods to elucidate the effect of pertinent parameters on electroosmosis in the patterned channel. The patterned microchannel results in an enhancement in the average EOF by creating an intrinsic velocity slip at the polyelectrolyte–liquid interface. An analytical solution of the EOF for a limiting case in which the groove width is much higher than the channel height is obtained based on the Debye–Huckel approximation. This analytical solution is in good agreement with the present numerical model when a low charge density and a thin Debye layer are considered. We have also established an analogy between the EOF in a polyelectrolyte-filled grooved-channel with the EOF in which the grooves are replaced by the charged slipping planes.

Heat Transfer Investigation of a Tube Partially Wrapped by Metal Porous Layer as a Potential Novel Tube for Air Cooled Heat Exchangers

Laminar forced convection heat transfer from a constant temperature tube wrapped fully or partially by a metal porous layer and subjected to a uniform air cross-flow is studied numerically. The main aim of this study is to consider the thermal performance of some innovative arrangements in which only certain parts of the tube are covered by metal foam. The combination of Navier–Stokes and Darcy–Brinkman–Forchheimer equations is applied to evaluate the flow field. Governing equations are solved using the finite volume SIMPLEC algorithm and the effects of key parameters such as Reynolds number, metal foam thermophysical properties, and porous layer thickness on the Nusselt number are investigated. The results show that using a tube which is fully wrapped by an external porous layer with high thermal conductivity, high Darcy number, and low drag coefficient, can provide a high heat transfer rate in the high Reynolds number laminar flow, increasing the Nusselt number almost as high as 16 times compared to a bare tube. The most important result of thisstudy is that by using some novel arrangements in which the tube is partially covered by the foam layer, the heat transfer rate can be increased at least 20% in comparison to the fully wrapped tube, while the weight and material usage can be considerably reduced.

Mixed convection inside a fluid‐porous composite cavity with centrally rotating cylinder

AbstractThe mixed convection in a fluid‐porous composite medium lying inside a square cavity with a centrally rotating cylinder has been investigated in the present work. The bottom half of the cavity is filled with a porous material and the top half is filled with a clear fluid. The bottom wall of the cavity is at a higher temperature, and the top wall is at a lower temperature. The vertical walls are thermally insulated. The convection inside the cavity sets through the combined mechanisms of the thermal buoyancy force and the shearing action of the centrally rotating cylinder. The relative importance of each driving mechanism over the other is featured through the Richardson number. The Darcy–Brinkman–Forchheimer equation is used for the flow modeling in the porous medium, and a single‐domain approach is adopted for the numerical solution in the fluid‐porous composite medium. The simulation is carried out with ANSYS Fluent software, and a parametric analysis involving the Rayleigh number (), Richardson number (), and the Darcy number () is conducted showing their effects on the flow and heat transfer. The phenomena are quite interesting at higher Darcy number and Rayleigh number. The distributions of isotherms, streamlines, and vector plots are plotted, along with the local Nusselt numbers for different parameters, to explore the underlying physics of the phenomenon. The system is found stable at lower Darcy number, and the heat transfer is minimum around Ri = 10. From the numerical study, an empirical correlation for the average Nusselt number is developed as a function of the other dimensionless numbers.

Continuous dependence for the Brinkman–Forchheimer fluid interfacing with a Darcy fluid in a bounded domain

This paper studies the continuous dependence of the Forchheimer coefficient λ and the Brinkman coefficient μ in a bounded domain of a viscous fluid interfacing with a porous solid. We assume that the viscous fluid is slow in Ω1, and the governing equations are Brinkman–Forchheimer equations. For the porous medium in Ω2, we suppose that the flow satisfies the Darcy equations. We can get the continuous dependence results of the solutions using the method of differential inequality.

Energy equality for the 3D critical convective Brinkman–Forchheimer equations

In this paper we give a simple proof of the existence of global-in-time smooth solutions for the convective Brinkman–Forchheimer equations (also called in the literature the tamed Navier–Stokes equations)∂tu−μΔu+(u⋅∇)u+∇p+αu+β|u|r−1u=0 on a 3D periodic domain, for values of the absorption exponent r larger than 3. Furthermore, we prove that global, regular solutions exist also for the critical value of exponent r=3, provided that the coefficients satisfy the relation 4μβ≥1. Additionally, we show that in the critical case every weak solution verifies the energy equality and hence is continuous into the phase space L2. As an application of this result we prove the existence of a strong global attractor, using the theory of evolutionary systems developed by Cheskidov.