Numerical Investigation of the Three-Dimensional Flow around a Surface-Mounted Rib and the Onset of Unsteadiness

Abstract

The incompressible laminar isothermal flow of a Newtonian fluid at steady state around a surface-mounted rib is studied in a three-dimensional (3D) numerical experiment. The dimensionless Navier–Stokes equations are solved numerically using the Galerkin finite element method for Reynolds numbers 1 to 800. The expansion ratio of the problem is 1:9.6, while the aspect ratio is 1:20. The transition from the steady to the unsteady state and the identification of the critical Reynolds number are investigated in this paper. Numerical results of the skin-friction lines at the bottom and streamlines throughout the computational field are presented. A comparison between the 2D and 3D flow is made to show the effect of the walls on the flow, which reaches the plane of symmetry and affects the flow there; hence, also affecting the stability of the flow. It is concluded that the flow is three-dimensional even for a Reynolds number equal to 10. The critical Reynolds number is 600, and the steady-state equations can be used for any calculations up to this value.

1. Introduction

Many researchers investigate flows around a construction in wind tunnels or water tunnels numerically or experimentally to solve and analyze problems related to agricultural engineering, such as natural ventilation, pressure and temperature distributions, air and water quality, heat transfer, humidity and CO₂ concentration, erosion, and many other factors associated with agricultural constructions, solar air heaters, pipelines, and the erosion of riverbeds and soils. Studying and solving problems in tunnels depends on the case under consideration and requires specialized knowledge of fluid mechanics and thermodynamics concerning fluid flows in tunnels, such as the effect of walls, conditions of instability and turbulence, similarity of conditions, etc.

Based οn the existing literature, the study of fluid flow around obstacles located in a wind tunnel mainly solves problems of airflow around greenhouses and livestock units, airflow through solar panels, and erosion of river bottoms and soils. Several numerical approaches have been used to simulate the crossflows through wind tunnels. Partheniotis et al. [1] studied turbulent flow over triangular and arched obstacles with the computer package ANSYS and vertically processed velocity profiles at different positions along the wind tunnel. Turbulent heat transfer was investigated by Meinders and Hanjalic [2] using two stacked surface obstacles in the form of cubes in a channel as well as other combinations of obstacles. Ntinas et al. [3] examined the two-dimensional air flow over various shapes of agricultural structures, both experimentally and computationally. Several investigators have worked with numerical methods, simulating the flow through a wind tunnel, and solving the governing equations. Thus, Iatan et al. [4] studied the flow of water over a coarse bottom in an open channel. They ensured the coarseness by using rows of hemispheric obstacles and used the DNS and LES method to solve the equations. Huang et al. [5] and Ikeda and Durbin [6], by the aid of DNS, calculated the friction during turbulent flow around rectangular obstacles that extended along the whole width of a channel. Sinha et al. [7] dealt with the flow of water over sediments in the shape of a hillock on river and channel beds. Singh and Debnath [8] numerically examined the flow of liquids over a coarse bed using stacked cubic obstacles. By applying the DNS method, Daniel et al. [9] numerically simulated the flow of air around sand hills for Reynolds numbers ranging between 500–3000. Kesmiri [10] studied the three-dimensional and two-dimensional flow of water in a channel with a coarse bed covered by two orthogonal obstacles along its width. Tauqeer et al. [11] numerically calculated the air flow over triagonal, quadratic, and hemispherical building constructions. Garron and Suresh [12] simulated the uniform flow of water through a channel for Reynolds numbers from 500 to 1500. They solved the governing flow equations with the finite-volume commercial FLUENT package. The finite-volume method has also been used to investigate the steady two-dimensional flow around an orthogonal-edged barrier by Boum et al. [13]. Direct numerical simulation has been used for constant, two-dimensional, viscous incompressible airflow over a bearing rib (Fragos et al. [14] 1997) or more than two consecutive obstacles (Ntinas et al. [1]). Malamataris [15] used direct numerical simulation to study the wall effects in a three-dimensional laminar flow over a backward-facing step with a constant aspect and expansion ratio. Hsieh et al. [16,17], using the SIMPLEC finite-difference method, solved the three-dimensional N–S equations around an orthogonal obstacle and presented the changes in the axial velocity u, the vertical velocity vector levels, and the line of equal velocity. Especially for the study of the effect of the walls and the unsteadiness, the researchers were concerned with similar flows around a step [18,19,20,21,22].

The current paper studies three-dimensional air flow around a surface-mounted rib on the wind tunnel by numerically solving the governing equations at steady state using FEM3D code [15,23,24,25,26,27]. This numerical experiment is considered a ‘primary’ investigation and aims to be a stepping-stone for further research on the nature of this flow. The term ‘primary’ is justified, as further research can be carried out based on the equations at steady state. The computational field discretization, the solved field equations, and the boundary conditions of the problem are presented in Section 2. In Section 3, the results of skin-friction lines or limiting streamlines on the bottom and streamlines are presented, and the question about when unsteadiness appeared in this numerical experiment is answered. Such results do not appear in other studies of flow around constructs, except in the study by Malamataris [15] for the three-dimensional laminar flow over a backward-facing step. In the present work, a comparison between two-dimensional and three-dimensional flow is also performed using FEM3D and 2D codes [14]. The critical value of the Reynolds number value is determined, above which the flow should be considered three-dimensional.

2. Materials and Methods

2.1. Computational Domain, Governing Equations, and Boundary Conditions

The computational domain of the steady three-dimensional flow around an orthogonal obstacle/construction can be seen in Figure 1. There are three directions of movement: x, the horizontal direction; y, the vertical; and z, the transverse. The dimensions are undifferentiated with the height, h, of the rib. The aspect ratio (ratio of rib height to wind tunnel height) is 1:9.6, and the expansion ratio (ratio of rib height to wind tunnel width) is 1:20, based on the Acharya et al. experiment [28] and the Fragos et al. numerical experiment [29].

The Navier–Stokes and continuity equations for isothermal and incompressible fluid flow used in the dimensionless form are the following:

$$\nabla ·u=0$$

(1)

$$\mathrm{u}·\nabla u=-\nabla p+\frac{1}{{Re}_h}\nabla ·\left\{\nabla u+{\left(\nabla \mathrm{u}\right)}^T\right\}$$

(2)

where u = (u,v,w) is the dimensionless velocity vector, dimensioned with the average velocity U₀ of the flow; p is the pressure dimensioned by the expression ρU²; and Re_h is the Reynolds number based on the height, h, of the rib and the velocity of the uniform flow U₀ as follows:

$${Re}_h=\frac{U_{o}h}{\nu }$$

(3)

where ν is the kinematic viscosity and ρ is the density of the fluid.

The boundary conditions for this flow are as follows:

At the entrance, that is, for x = 0, 0 < y < 9.6, 0 < z < 10:

w = 0

(4)

v = 0

(5)

u = (−6/92.1)∙y∙(y − 9.6)

(6)

At the bottom, the top, the side walls and the surface of the rib, and with no slip conditions:

w = 0

(7)

v = 0

(8)

u = 0

(9)

At the symmetry level:

w = 0

(10)

∂u/∂z = 0

(11)

∂v/∂z = 0

(12)

At the exit of the calculational field, there are free boundary conditions.

At the entrance of the computational field, Equations (4)–(6) impose a parabolic-slit flow up to the lateral wall.

The symmetry plane is subject to the symmetric condition w = 0, while a non-slip condition is imposed on the lateral walls, bottom, and rib. At the exit of the computational field, a free boundary condition is imposed in order to let the fluid leave the computational domain freely without any distortion of the flow in the interior.

2.2. Finite Element Formulation and Computational Code Used

For the solution of the governing equations, the Fortran code to implement the computations is called FEM3D [15,24,25,26,27], which is a code written in-house. In the present work, modifications were made to simulate the flow around the rib. The FEM3D code used in this study has been tested and validated successfully in many recent publications dealing with three-dimensional flows [15,23,24,25,26,27] and was subsequently run through the ARIS supercomputer system. A computational grid was used to encapsulate hexahedral cuboid elements with 27 velocity knots and the 8 vertex nodes at which the pressure was calculated. The computational field elements are shown in

Table 1. Discretization of the computational domain.

Items	Values
Elements in x-direction, before the rib	33
Elements in x-direction, after the rib	101
Elements in x-direction, above the rib	28
Elements in y-direction	62
Elements in z-direction	45
Total number of nodes	185,406
Total number of unknowns	4,571,526
Maximum front width	22,567
Computing time per iteration (256 cores)	7.5 h
Initial Reynolds number	0.02
Maximum error for velocity	6						10−6
Maximum error for pressure	6						10−4
x-coordinates of the vertices	0	1	2	3	4	5	6	7	8	9
	10	10.5	11	11.5	12	12.5	13	13.2	13.4	13.5
	13.55	13.6	13.65	13.75	13.8	13.85	13.9	13.95	13.96	13.97
	13.98	13.99	14	14.01	14.02	14.03	14.04	14.05	14.1	14.15
	14.2	14.25	14.3	14.35	14.4	14.45	14.5	14.55	14.6	14.65
	14.7	14.75	14.8	14.85	14.9	14.95	14.96	14.97	14.98	14.99
	15	15.01	15.02	15.03	15.04	15.05	15.1	15.15	15.2	15.25
	15.3	15.35	15.4	15.45	15.5	15.6	15.7	15.8	15.9	16
	16.1	16.2	16.3	16.4	16.5	16.6	16.7	16.8	16.9	17
	17.2	17.4	17.6	17.8	18	18.2	18.4	18.6	18.8	19
	20	21	22	23	24	25	26	27	28	29
	30	31	32	33	34	35	36	37	38	39
	40	41	42	43	44	46	47	48	49	50
	51	52	53	54	55	56	57	58	59	60
y-coordinates of the vertices	0	0.01	0.035	0.06	0.085	0.11	0.14	0.2	0.3	0.4
	0.5	0.6	0.7	0.8	0.85	0.9	0.95	0.97	0.98	0.99
	1	1.005	1.01	1.02	1.03	1.04	1.05	1.06	1.1	1.25
	1.5	1.75	2	2.33	2.67	3	4	5	6	7
	8	8.5	8.75	9	9.1	9.2	9.3	9.4	9.5	9.55	9.6
z-coordinates of the vertices	0	0.25	0.5	0.75	1	1.5	2	2.5	3	3.5
	4	4.5	5	5.25	5.5	5.75	6	6.25	6.5	6.75
	7	7.25	7.5	7.75	8	8.25	8.5	8.75	9	9.25
	9.365	9.5	9.55	9.6	9.65	9.7	9.75	9.8	9.85	9.9
	9.95	9.975	9.9875	9.99375	9.996875	10

. First, the finite Galerkin method is applied to discriminate the field; then, the unknown velocities and pressures are given relative to their values at the nodes of each element in the relationships:

$$u=\sum _{i=1}^{27}{u_i\phi }^i\left(a\right),v=\sum _{i=1}^{27}{v_i\phi }^i\left(b\right),w=\sum _{i=1}^{27}{w_i\phi }^i\left(c\right),p=\sum _{i=1}^8{p_i\psi }^i(d)$$

(13)

where φⁱ and ψⁱ are the Lagrangian tri-quadratic and tri-linear basis functions, respectively. The discrete lattice around the rib is shown in Figure 2.

Equations (14) and (15) give the following weighted balances for continuity and motions, respectively:

$$R_C^i={\int }_V^{}\nabla ·u{\psi }^{i}dV$$

(14)

$$\begin{gathered} R_C^i={\int }_V^{}\left[\mathrm{u}·\nabla u{\phi }^i-\left(-pI+\frac{1}{{Re}_h}\left\{\nabla u+{\left(\nabla \mathrm{u}\right)}^T\right\}\right)·\nabla \phi \right]dV \\ -{\int }_S^{}n·\left[-pI+\frac{1}{{Re}_h}\left\{\nabla u+{\left(\nabla \mathrm{u}\right)}^T\right\}\right]{\phi }^{i}dS \end{gathered}$$

(15)

Claiming that these weighted balances stretch to zero, we obtain a system of n algebraic non-linear equations with the n-node unknowns solved by the iterative Newton–Raphson method. The code converges after 4–5 iterations for each Re number, starting with Re_h = 0.02 and reaching up to Re_h = 1425. The duration of each iteration reached 7.5 h. Further mesh densification showed no significant change in the numerical solution. Additionally, for the comparison between 3D and 2D flow, two-dimensional computations have been made with the computational code for the three-dimensional flow using one element in the z-direction.

3. Results and Discussion

3.1. Data Analysis

The purpose of this paper is to analyze data derived from the execution of the FEM3D code through the ARIS supercomputer system in order to draw useful conclusions for researchers performing numerical and laboratory experiments in wind and water tunnels for agricultural applications.

3.1.1. Two-Dimensional vs. Three-Dimensional Simulation

Figure 3 shows the flow lines downstream of the rib in a two-dimensional and three-dimensional approach for Reynolds numbers from 10 to 500. Ιt is observed that the vortices downstream of the rib have qualitative and quantitative differences in the two flows, even for Re = 10. Their qualitative difference is that in the two-dimensional flow, the streamlines are well-formed closed curves, while the streamlines do not appear to take on a clean closed-curve shape for the three-dimensional flow. The same phenomenon is observed for Re = 100 to 500. According to Malamataris 2013 [15], this suggests that the flow can be considered three-dimensional even at the plane of symmetry. Therefore, the two-dimensional equations do not accurately describe the downstream flow for Re number greater than 10. Their quantitative difference is that vortices in the three-dimensional flow have a longer recirculation length than those in the two-dimensional flow. The increased downstream recirculation length in the three-dimensional flow in relation to that of the two-dimensional with the increase in the Reynolds number is shown in Figure 4. In the three-dimensional flow, this probably results from the effect of the walls. Therefore, the streamlines are directed transversely to the symmetry plane, increasing the size of the downstream vortex. In Figure 5, the length of the upstream vortex is shown in the plane of symmetry for two-dimensional and three-dimensional flow and for Reynolds numbers 25 to 500. It is observed that these vortices are almost similar, both qualitatively and quantitatively, in both flows. The size of the vortices changes minimally with the increase in the Reynolds number, as shown in Figure 5 and Figure 6.

3.1.2. Three-Dimensional Streamline Patterns Downstream of the Rib

In Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13, the streamlines downstream of the rib can be seen at selected vertical levels—so that the flow field can be better visualized—and from a height of y = 2 h up to y = 9.6 h, which is the height of the roof, since it was determined that beyond that height, no significant change to the flux characteristics is evident, and any research-worthy points are observed at approximately y = 2 h.

The figures show the increase in the length of recirculation relative to the Re number and the effect of the wall on the flow as we approach the lateral wall.

The effect of the lateral sides of the wall is equivalent to the change in the quality of the streamline patterns relative to the usual ones. It is also observed that with an increase in the number or Re, the effect of the wall moves toward the symmetry plane downstream of the rib; similarly, the length of the recirculation area increases with the Re number. Therefore, it is not necessary to study the flow lines for Re > 600 based on the current constant flow model, since another model for unstable or turbulent situations will be needed in this case.

3.1.3. Βottom Friction Lines Downstream of the Rib

Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22 and Figure 23 show the skin-friction lines on the bottom, downstream of the rib, for Re numbers from 1 to 800. As pointed out in the introduction, the main purpose of this work was to determine where, when, and why the steady flow stops and the flow becomes unsteady. For these questions to be answered, the equality friction lines on the bottom coinciding with the limit flow lines of the bottom in three-dimensional flow [12,17] had to be constructed. Therefore, according to pre-existing studies and for flows around a step [12,14,15], the change of direction of the reattachment line due to unsteadiness would not be studied in 2D flows due to the absence of lateral walls. At large Reynolds numbers and for three-dimensional flow around a rib, the unsteadiness starts when the flow lines are directed from the symmetry level toward the side walls, and the the shape of the reattachment line (the geometrical locus of the reattachment points downward of the rib) loses its regularity and somehow becomes curved. The specific behavior of the flow is due to the effect of the lateral walls on which the fluid impinges and is pushed back due to the non-slip boundary conditions and the plane of symmetry. In the two-dimensional flow, this specific behavior is not observed due to the absence of lateral walls. A similar behavior of the flow was found in the flow over a backward-facing step by Malamataris in 2013 [15].

As shown in Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22 and Figure 23, the limiting streamlines of the fluid are directed toward the symmetry plane and the reattachment line is smoothly sloped. In Figure 15, for Re = 600, the reattachment one is bent and loses its smoothness, while the boundary line is directed from the symmetry plane toward the side wall. At this point, it can be safely deduced that the boundary for which unsteadiness first occurs is Re = 600. It is noted that the observed unsteadiness is not due to the instability of the code, but to the flow itself. Convergence for Re numbers greater than 600 is difficult to reach, and often, the code diverges. In these cases, an unsteady-state code would better approximate the flow. In Figure 9 and Figure 16, i.e., for Re = 700, the flow appears to be completely unsteady.

As previously stated, the reattachment lines can be clearly seen from the friction lines on the bottom.

4. Conclusions

In this study, an attempt was made to determine for up to what velocity or Reynolds number we can safely state that the flow is uniform and when unsteadiness begins for the flow around a rib mounted in a wind tunnel by directly solving the Navier–Stokes and continuity equations. Several conclusions were drawn from the analysis of the results of the numerical experiments, the most important of which can be mentioned as follows:

• Even for Re = 10, the flow can be considered 3D downstream of the rib.
• The vortices downstream of the rib show qualitative and quantitative differences between two-dimensional and three-dimensional flows with increasing Re numbers.
• Upstream vortices show no differences and remain almost constant with increasing Re numbers.
• The results of limiting streamlines or equal shear stress on the bottom of the wind tunnel show that the unsteadiness begins at a Reynolds number relative to a height of the rib equal to 600.
• The steady-state model should be used for Re_h values up to 600; for values greater than 600, the flow becomes unstable and turbulent, and a code utilizing unsteady flow equations is recommended.
• The skin-friction lines on the bottom show the points of reattachment of the flow to the recirculation area downstream of the rib. The increase in the recirculation length with the increase in the Reynolds number is clearly apparent.
• The streamlines at a vertical level downstream of the rib confirm our findings on the limiting streamlines, especially near the walls, due to the wall effect on the flow.
• An analysis should precede the steady-state study so that the exact transition point is identified. The results showed that beyond this point, an unsteady-flow model is needed due to the occurred unsteadiness.