Computational Study of the Flow of Newtonian Fluid Through A Straight Channel and Lid-Driven Cavity

This article aims to introduce a numerical study of two different incompressible Newtonian fluid flows. The first type of flow is through the straight channel, while the second flow is enclosed within a square cavity and the fluid is moved by the upper plate at a specific velocity. Numerically, a Taylor-Galerkin\ pressure-correction finite element method (TGPCFEM) is chosen to address the relevant governing equations. The Naiver-Stoke partial differential equations are usually used to describe the activity of fluids. These equations consist of the continuity equation (conservation of mass) and the time-dependent conservation of momentum, which are preserved in Cartesian coordinates. In this study, the effect of Reynolds number ( Re ) variation is presented for both problems. Here, the influence of Re on the convergence rate and solution behavior is provided. Findings display that, there is a significant impact of Re upon the temporal convergence rates of velocity and pressure. As well, the rate of convergence increases as the values of Re are risen. For the cavity problem, one can infer that, as the Reynolds number rises, the size of the vortex is reduced


1.Introduction
The numerical analysis of differential equations that govern the incompressible Newtonian fluid flow through straight channel and lid-driven cavity has received some great interest in the field of fluid dynamics. These types of fluid are usually governed by two differential equations; called conservation of mass and time-dependent conservation of momentum, which are offered here in cartesian coordinates under isothermal conditions (see Bird et al. 1987 for details). Geometrically, these problems appear at first sight to be simple, but it has a very important role in the field of fluid research in various industries. In the first problem, Poiseuille (Ps) flow along a two-dimensional planar straight channel, under isothermal condition is provided. In contrast, the lid square driven cavity problem is presented in this study as well. This flow is an idealized representation of several engineering situations, such as the flow over cutouts and repeated slots on the walls of heat exchangers or on the surface of aircraft bodies. In addition, as it is well known there are several structures of the lid-driven cavity depending upon boundary conditions, for example, single wall driven, parallel wall driven and anti-parallel wall driven etc. [1]. Here, we concern with the single upper wall driven, where the fluid is motioned by the upper-plate (lid) at a specific velocity. This problem has been investigated quite extensively as a numerical normative problem and as a test base for studying specific physical impacts, see [2]. Several numerical studies have been conducted and extended to steady incompressible flow in a driven cavity and are covered in the literature [3][4][5][6][7][8].
Numerically, a semi-implied Taylor-Galerkin/pressure-correcting finite element method (TG/PC-FEM) is applied in the temporal gradient. This approach is introduced by Townsend and Webster [9] to address incompressible flows of Newtonian and non-Newtonian fluids. In this method, the variables of velocity and pressure are decomposed (known primitive variable formulations), and this idea was inspired by Corinne's investigation [10]. Under this scheme, two main directions were taken to treat the governing equations. The first direction includes the fractal step method, which is favored by Gresho et. al. [11] and Donea et. al. [12]. Also, based on the velocity correction approach another direction is taken to solve the governing flow equations (see Kawahara and Ohmiya [14]). In addition, this approach includes two phases, the first being the Taylor-Galerkin method, which is a two-step Lax Wendroff time-step action (prediction corrector), extractor via Taylor series expansion in time (Donea [16], Zinenkiewicz et al. [17]). The second phase is the pressure correction method accommodate the incompressibility constraint to assure second-order thoroughness in time (see Hawken et al. [18], Aboubacar et al. [19]). The scheme is applied to triangular FE meshes, with pressure nodes located at the vertices and velocity components at both the vertical points and the middle nodes. Amazing attention was spent to treat the flow problems in the cartesian coordinates by using TS-TG/PC-M , see for example [15], [18][19][20].
The main motivation of this present study is the treatment of flow problems in Cartesian coordinates using Taylor-Galerkin/ pressure correction approach. In this context, two benchmark problems are studied; Poiseuille flow along a planar channel and lid-driven cavity of a square domain. The main findings in both problems are focused on the temporal convergence rate of the system solution to be a steady state under the variation of the Reynolds number. The mathematical modelling of the motion of the Newtonian flows is present in the next section. The numerical approach is given in section 3. The problem of the discretization, the boundary conditions and the related numerical results are presented in sections 4 and 5, respectively.

Mathematical Modelling
For incompressible Newtonian isothermal flow in state of absence of body forces, the governing equations, which consist of momentum and continuity equations, can be stated as: where, u, p, ρ and d are the velocity, pressure, density and rate of deformation for general flows, which can be expressed as Further, to define these equations in non-dimensional form, we introduce the scales and nondimensional variables for the length L (the exact half-channel width), the velocity (the exact average velocity), pressure and viscosity as follows: In addition, Re= represents the number of the non-dimensional set of Reynolds numbers.
Regardless of the asterisk for clarity, the corresponding system of equations is shown under isothermal conditions (4) In the Cartesian components, these equations can be written as: Where ux, uy and uz represent the velocity in x direction, y direction and z direction, respectively.

Numerical method
A time semi-implied Taylor-Galerkin/ pressure-correction numerical scheme was adopted in this study. This scheme is based on a partial-staged approach that is first defined in the time domain by Taylor series expansions and includes a two-step Lax-Wendroff approach to capture second-order precision in time. Subsequently, the specific modification of the actuator split technology which is called the pressure correction method is adopted. This technique was developed by (Van Kan, 1986) which provides a second degree of correctness and overall strength by means of linear energy analysis. Historically, these schemes were proposed by Hawken et al., in 1990 in explicit form and then recognized by Townsend and Webster in 1987 who created a scheme which is named Taylor-Galerkin/ Pressure-correction (TGPC). This algorithm can be presented in three distinct fractional stages for each time step as follows: In summary, the algorithm consists of three stages over each time [ , +1 ]. First, at stage 1a, velocity components at half-time step ( + ). In stage 2, the pressure difference equation is solved over the full-time step range. Finally, the velocity field is recovered at stage 3, to complete the time step cycle. To complete the process of the work, we need to get the matrix for the above stages by applying the essential steps in the finite element method. In this context, after finding the weak formulation of the stages ((9a), (9b), (9c) and (9d)), we introduce approximations ( , ) and ( , ) to the velocity and pressure respectively over finite dimensional function spaces as follows: ( , ) = ∑ =1 ( ) ( ), Such that and are the number total of the nodes and the number of vertices of the triangles only, respectively. Here, ( ) and ( ) represents a vector of nodal values of velocity and pressure and ( ), ( ) are their respective basis (shape or interpolation) functions. In this context, quadratic shape functions of the velocity components in Cartesian coordinates are used. These functions are given in the usual coordinates as: (12) In contrast, for pressure, the following linear shape function is used: Here, the vector of the linear shape function is symbolized by [E].
Where, is the area of the triangle of the element and , and are coefficients. Thus, the corresponding a TGPC form of equations ((9a), (9b), (9c) and (9d)) can be written in matrix form as , see [18]:

Test Problems and Boundary conditions
To confirm the algorithm, we review two problems, regarded as benchmarks in the Newtonian regime of interest. The first problem is the flow of Poiseuille through a straight channel with two axial dimensions under isothermal conditions. For this purpose, four triangular finite elements mesh are implemented: M1(4×4), M2(8×8), M3(14×12), M4(40×40) with the same length, as shown in Figure 1(a-d) (characteristics of a typical finite element mesh are included in Table 1). A second example concerns a driven cavity problem, where the fluid is driven by the upper plate (lid) at a given velocity as illustrated in Figure 1e. Here, triangular finite elements mesh is used as well; M5(10×10).  (1) Poiseuille flow( ) at the entrance is specified with zero radial velocity.
(2) Non-slip is applied to the upper and lower walls of the channels. (3) Zero radial velocity is applied and zero pressure is applied to the outlet of the channels.

(b) Driven cavity problem:
(1) At the upper boundary, the tangential velocity with a variable profile of type = 16 2 (1 − ) 2 is applied to pay the fluid flow in the cavity with a variable profile of type. (2) No-slip is applied on the remaining three walls.

Numerical results
In the present study, two numerical examples are considered to calibrate the algorithm; a two-dimensional axisymmetric straight channel and driven cavity. Here, the numerical results of Newtonian flow are concerned on the effect of the Reynolds number ( ) on the solution components. The essential findings focus on the rate of convergence of velocity and pressure solutions under variation of Reynolds number. For that purpose, a time fractional-staged Taylor Galerkin/ pressure correction (TGPC) framework is applied. The convergence criteria are taken here as 10 −6 and the typical Δ is (10 −2 ). Al the results are introduced for the Cartesian coordinates system.

Straight channel
In Figure 3, velocity and pressure fields for fine mesh are presented at Re=1. As expected, the maximum axial velocity level is displayed along the center line of the channel with maxima of around (u=1.01791), and then gradually decreases. Also, the maximum level of the pressure (p=9.4244 units) has appeared at the inlet of the pipe decreases as we approach the outlet of the pipe. In addition, more details of the results for each mesh are in Table 2.  Pressure drop is plotted in Figure (4) for four different meshes and three different Reynolds number Re={1,5,10}. The results reveal that, a linear pressure drop occurs throughout the center of the channel, after which the pressure reaches zero at the channel port. Additionally, in the case of Re-variation, the level of pressure rises with increasing Re, which reaches the highest level of 24 units with Re=10. Also, for the level of pressure, one can see that there is no change in the pressure when we refine the mesh. The velocity gradients will be developed by increasing the Reynolds number, which gives the difficulties of convergence for the large Re numbers. Therefore, we devoted our attention to studying the impact of this factor on the level of velocity convergence. Figure (5) shows the rate of convergence of the axial velocity through the variation in the Reynolds number ( ) for four different meshes. The result reflects the effect of Re-variation on the axial velocity convergence, where the convergence rate increases corresponding to the increase in Re-level. Again, for the different meshes the level of convergence increases with the increase in mesh accuracy by a small amount. Abdul-Jabbar et al. Iraqi Journal of Science, 2023, Vol. 64, No. 8, pp: 4043-4057 4051 The convergent pressure is provided in Figure (6) for four different meshes through the variation in Reynolds number ( ). Here, the same feature of the convergent velocity is observed in the pressure case, where the level of convergence is increased with increasinglevel. Here, for the difference meshes no obvious change in the level of convergence appeared.

Driven cavity
Generally, this example is used as a standard test for evaluating numerical methods for incompressible flow (see Ghia et al., 1982, Hawken et al., 1990. In this problem, the flow is enclosed within a square cavity and the fluid is moved by the upper plate at a specific velocity. The velocity distribution in the axial direction is shown in Figure 7(a) at fixed Re=1. For the field, one can observe that, while there is a large gradient near the lid and dead zones in the lower left and lower right corners. Moreover, the pressure field is illustrated in Figure 7  (a-f) shows streamline patterns of velocity flow through the variation in Reynolds number ( ). Here, we have been focused on the steady state vortex behavior as a function of rising Re. Generally, vortex size reduction is clearly apparent with increasing Re. In this context, for a Reynolds number less than 10, the vortex size at the cavity center is larger than that with Re≥10 due to the energy squander. Also, low Reynolds numbers, secedes the flow near the lower right and left corners and here two vortices formed. We can also see if the Reynolds number is raised, then there is more deadlock in flow, which causes her secession along the wall earlier and create bigger angle vortices. By increasing Reynolds number, vortices form in the down corners as well as one in the upper left corner.  Science, 2023, Vol. 64, No. 8, pp: 4043-4057 4054 As stated above, the convergence rate of the axial velocity and pressure are shown in Figure  ( 9) for different values of Reynolds number (Re = 1, 5, 10). In general, the level of velocity convergence at the same level of pressure convergence due to the effect of the nonlinear behavior. Additionally, and where expected, the results indicate that the rate of convergence increases with increasing Re values due to compression effects

Conclusion
In this study, the numerical simulation for two different incompressible Newtonian fluid flows is achieved. The first type of flow is through the straight channel, while the second flow is enclosed within a square cavity and the fluid is moved by the upper plate at a specific velocity. To address the governing equations, the Taylor-Galerkin/ pressure-correction finite element method (TGPCFEM) is chosen. For that purpose, five finite element meshes are utilized. Convergence analysis is performed for the velocity and the pressure under the variation of Reynold's number for both problems. In this context, the results detect that there is a significant impact of Re upon the temporal convergence rates of velocity and pressure. Additionally, the rate of convergence is increasing as the values of Re are risen. As for the cavity problem, the results show that when the Reynolds number rises, the size of the vortex decreases and vortices form in the down corners as well as one in the upper left corner.