Analysis of flow over backward facing step with transition

Abstract

The present study deals with the study of the velocity distribution and the separation phenomenon of flow of air over a two dimensional backward facing step. The flow of air over a backward facing step has been investigated numerically using FLUENT. Flow simulation has been carried out in a backward facing step having an expansion ratio (ratio of the height before and after the step) of 1:1.94 and the results obtained are compared with the published experimental results. Comparison of flow characteristics between steps with three different transitions is made. The variation of reattachment length for all the three cases are analyzed for wide range of Reynolds number ranging from 100 to 7000 which covers the laminar, transition and turbulent flow of air. Simulation of the flow over steps with expansion ratios of 1:1.24, 1:1.38, 1:1.47, 1:1.53, 1:1.94, 1:2.20 are also carried out to examine the effect of different expansion ratios on the reattachment length. It is found that the primary reattachment length increases with increase in the expansion ratio. The primary reattachment length at the bottom wall downstream of the step is minimum for the step with round edged transition and maximum for the step with a vertical drop transition.

Keywords

Reattachment length / backward facing step / transition / flow separation / k-ϵ model

Introduction

The complex behavior of fluid flow in the region of separation and subsequent reattachment is commonly encountered in many engineering applications. Flow separation occurs in aerodynamic devices such as high lift air foils at large angles of attack, turbines and compressor blades, diffusers, flow around the buildings, suddenly expansion of pipes etc. Transition in pipes frequently occurs in pipe networks where the flow separation takes place. Sudden expansion in pipes involves considerable head loss due to the boundary layer separation. Flow around bluff bodies typically separates resulting in shedding of vortices that impact the body with some periodicity [ 1]. From aerodynamic perspective, the major reason for consumption of maximum fuel is the drag force which is created due to flow separation. Also the understanding of the fluid structure interaction at the separated shear layer instabilities is necessary to design the structure. To study the region of the separated flows, the backward facing step is considered due to the simplicity in geometry, its single fixed separation point and little or no effect on the wake dynamics by the downstream disturbances. A number of researchers conducted the experiments on flow of air over a backward facing step and measured the velocity distribution, reattachment length and the recirculation zone for a wide range of Reynolds number [ 2– 8]. The experimental results show that the various flow regimes are characterized by typical variations of separation length with Reynolds number [ 2]. Also an additional region of flow separation at the downstream of the step is found apart from the primary recirculation region [ 2]. The study of the turbulent flow in the downstream of a backward facing step shows the length of the separation zone is about five times the step height [ 3]. The investigations of the size of the reverse flow regions with the increase of the Reynolds number from 100 to 8000 shows that the size of the reverse flow regions is a function of Reynolds number [ 8]. The reverse flow regions increases and moves further downstream in case of laminar flow, the size of reverse flow regions decreases and moves upstream for the transition flow and for the turbulent flow, the size of reverse flow regions remains constant with increase of the Reynolds number [ 8]. Several correlation equations have been developed to predict the reattachment length of the recirculation region [ 9]. The Reynolds stress is found to have the maximum value upstream of the reattachment [ 10]. The study on effect of the inlet boundary conditions on the flow over a backward facing step shows the blunt edge will result in higher turbulent intensity upstream of the step as it will introduce additional turbulence into the turbulent boundary layer upstream of the step height compared with a sharp leading edge [ 4]. Also, number of researchers simulated numerically the flow over a backward facing step and analysis is done for a wide range of Reynolds number [ 6, 10– 21]. Several numerical models have been developed to predict the reattachment length. Non equilibrium wall function with modified k-ϵ model predicted the closest reattachment length [ 22]. Three dimensional computational stability analysis of flow over a backward facing step shows that the first absolute linear instability of the steady two dimensional flow is a steady three dimensional bifurcation at a critical Reynolds number of 748 [ 6]. The flow past a backward-facing step of expansion ratio two can exhibit large transient growth at Reynolds numbers well below that for asymptotic instability, and that this growth can be predicted within the framework of linear optimal perturbations, which provide theoretical underpinning for the largely phenomenological study of local convective instability in previous investigations of backward-facing step flows [ 12, 21]. Two dimensional numerical simulations of this step geometry underestimate the experimentally determined extent of the primary separation region for Reynolds number greater than 400 and this disagreement between physical and computational experiments is due to the onset of three dimensional flow near Reynolds number 400 [ 14].

Very few literatures are found on the analysis of flow over a step having different expansion ratio as well as steps with transition such as step with rounded edge. But almost no analysis is available regarding recirculation region on step with certain inclination. The turbulent flow over step with rounded edges shows a relatively shorter separation bubble and a slower return to the developed flow condition in comparison to the step without transition [ 6]. The primary reattachment length increases with the increase in step height [ 17]. The recirculation length increases nonlinearly with increasing expansion ratio [ 23]. Inside the primary recirculation flow region, the maximum friction coefficient develops along the centerline of the duct and its magnitude increases as the step height increases while its position moves further downstream as the step height increases. But downstream from the reattachment line and outside the primary recirculation flow region, the friction coefficient along the centerline of the duct decreases with increasing step height [ 17]. In that region the maximum friction coefficient develops close to the sidewall. The presence of large scale vortex structures was found even after the reattachment length [ 7]. The analysis of pressure losses for various expansion ratios and Reynolds numbers shows the increase in the pressure losses with increase in step height and decrease with increase in Reynolds number [ 23]. The transitional regime flow over a backward facing step show that the flow becomes unsteady and exhibit three dimensional behavior [ 20].

But in spite of all these efforts, there is a definite dearth of information regarding the turbulent flow structure both downstream and upstream of a backward facing step with gradual transitions such as step with curved edge as well as step with different inclination. Almost no literature is available regarding the analysis of recirculation zone in step having inclined transition as well as rounded edge. So the present work is carried out to add the existing knowledge of backward facing step flow with gradual transitions and to deepen the understanding of internal flow with separation over a wide range of Reynolds number covering laminar, transition and turbulent flows. The objective of this study is to examine the flow characteristics over a vertical, curved and inclined backward facing step.

Governing equations

The governing equations considered for the analysis of steady two-dimensional turbulent flow over backward facing step can be expressed as below [ 2]:

Continuity equation

(1) {\displaystyle \frac{\partial \left(\mathit{\rho }u\right)}{\partial x}}+{\displaystyle \frac{\partial \left(\mathit{\rho }v\right)}{\partial y}}=\mathrm{0.}

Momentum equation

(2) {\displaystyle \frac{Du}{Dt}}=X-{\displaystyle \frac{\mathrm{1}}{\mathit{\rho }}}{\displaystyle \frac{\partial p}{\partial x}}+{\displaystyle \frac{\mathit{\mu }}{\mathit{\rho }}}\left({\displaystyle \frac{{\partial }^{\mathrm{2}}u}{\partial x^{\mathrm{2}}}}+{\displaystyle \frac{{\partial }^{\mathrm{2}}u}{\partial y^{\mathrm{2}}}}\right){\displaystyle \frac{Dv}{Dt}}=Y-{\displaystyle \frac{\mathrm{1}}{\mathit{\rho }}}{\displaystyle \frac{\partial p}{\partial y}}+{\displaystyle \frac{\mathit{\mu }}{\mathit{\rho }}}\left({\displaystyle \frac{{\partial }^{\mathrm{2}}v}{\partial x^{\mathrm{2}}}}+{\displaystyle \frac{{\partial }^{\mathrm{2}}v}{\partial y^{\mathrm{2}}}}\right),

where r is mass density, µ is dynamic viscosity, u and v are the velocity component in x and y direction respectively. X and Y are the body force in x and y direction respectively. The standard k-ϵ turbulence model is used for flow calculation due to its robustness and reasonable accuracy for a wide range of turbulent flows which can be expressed as below [ 24]:

k-equation

(3a) {\displaystyle \frac{\partial \left(\mathit{\rho }k\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\mathit{\rho }ku\right)}{\partial x}}={\displaystyle \frac{\partial }{\partial x}}\left[\left(\mathit{\mu }+{\displaystyle \frac{{\mathit{\mu }}_t}{{\mathit{\rho }}_k}}\right){\displaystyle \frac{\partial k}{\partial x}}\right]+G_k+G_b-\mathit{\rho }\mathit{ϵ}-Y_m+S_k\mathrm{.}

ϵ-equation

(3b) {\displaystyle \frac{\partial \left(\mathit{\rho }\mathit{ϵ}\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\mathit{\rho }\mathit{ϵ}u\right)}{\partial x}}={\displaystyle \frac{\partial }{\partial x}}\left[\left(\mathit{\mu }+{\displaystyle \frac{{\mathit{\mu }}_t}{{\sigma }_{\mathit{ϵ}}}}\right){\displaystyle \frac{\partial \mathit{ϵ}}{\partial x}}\right]+C_{\mathit{1}\mathit{ϵ}}{\displaystyle \frac{\mathit{ϵ}}{k}}\left(G_k+C_{\mathrm{3}\mathit{ϵ}}{G}_b\right)-C_{\mathrm{2}\mathit{ϵ}}\mathit{\rho }{\displaystyle \frac{{\mathit{ϵ}}^{\mathrm{2}}}{k}}+S_{\mathit{ϵ}}\mathrm{.}

where k is turbulent kinetic energy, ϵ is rate of dissipation, G_k is turbulence kinetic energy due to the mean velocity gradient, G_b is turbulence kinetic energy due to the buoyancy, Y_m is contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, S_k and S_ϵ are source terms, s_k and s_ϵ are the turbulent prandtl numbers for k and ϵ respectively and the values are s_k = 1.0 and s_e = 1.3 respectively, C_1ϵ, C_2ϵ and C_3ϵ, are constants and the values are C_1ϵ = 1.44 and C_2ϵ = 1.92.

(4) C_{\mathrm{3}\mathit{ϵ}}=\tanh \left|{\displaystyle \frac{v}{u}}\right|,

where v is the component of flow velocity parallel to the gravitational vector and u is the component of flow velocity perpendicular to the gravitational vector. C_3e = 1 for buoyant shear layers for which the main flow direction is aligned with the direction of gravity and C_3e = 0 for buoyant shear layers that are perpendicular to the gravitational vector.

The turbulent or eddy viscosity μ_t is computed by

(5) {\mathit{\mu }}_t=\mathit{\rho }{C}_{\mathit{\mu }}{\displaystyle \frac{k^{\mathrm{2}}}{\mathit{ϵ}}},

where C_μ = constant and the value is 0.09.

Boundary conditions

The boundary conditions employed for the solution are as follows:

At walls: u = v = 0; inlet: u = u and v = 0; exit: at x = l, {\displaystyle \frac{\partial u}{\partial x}}=\mathrm{0} and {\displaystyle \frac{\partial v}{\partial x}}=\mathrm{0} .

The solution of the above partial differential equations is obtained using FLUENT. Finite volume scheme is used for numerical discretization. The solution procedure starts by assuming an initial guess value and then computes a converged solution by iteration.

Computational approach using FLUENT

For the present investigation, a sudden expansion duct with an expansion ratio of 1:1.94 is considered. The step height is 4.9 cm. The height of downstream of the step is of 10.1 cm. Inlet channel length is kept 20 cm, whereas the length of duct is 80 cm from the step. The significant length of outlet channel ensures that the outlet condition does not affect the flow near the step. Figure 1 shows the geometry of the backward facing step. During the flow over the backward facing step, two recirculation regions are observed, one at the bottom wall and the other at the top wall. The location of both the recirculation region is shown in a schematic in Fig. 1. The primary separation of flow occurs at the point of the expansion at the bottom wall and an additional separation region occurs in the upper wall at a distance of x₂ from the step. Reattachment of separated boundary layer takes place at a distance of x₁ from the step for the bottom wall and at a distance of x₃ from the step for the top wall. Figure 2 shows the grid for the vertical, round edged and inclined backward facing steps. Rectangular grid is created for vertical step and triangular grid is created for curved and inclined steps. The solution is obtained by FLUENT using SIMPLE algorithm. First order upwind scheme is used to discretize the momentum equation. Laminar model is used for the solution of laminar flow and k-ϵ turbulent model is used for the simulation of turbulent flow. Standard wall function is used as it offers significant reduction in computational expense compared to low Reynolds number formulation. Pressure based solver is used since the flow is incompressible and separation occurs due to adverse pressure gradient. Table 1 shows the reference values of different parameters used in this study and Table 2 shows the boundary conditions of different zones used in this study. The discretization schemes used for various equations used in the simulation are shown in Table 3. Table 4 shows the relaxation factors of different variables used in this study. A convergence criterion of 1 × 10^-3 is provided for continuity equation, x-velocity, y velocity, turbulent kinetic energy and turbulent dissipation rate. Initially the domain is discretized into 900 rectangular grids uniformly and the reattachment length is determined for Reynolds no 1000 and 10000 which cover both laminar and turbulent flow. The result is compared with finer grid to satisfy the grid independency. Initially all the 900 grids are refined so that the total number of grid becomes 3600 and the value of the reattachment length for Reynolds number 1000 and 10000 is determined. Also the grids only at the boundary are refined so that the total number of grid becomes 2100 and the value of reattachment length is determined. The value of reattachment length for both uniform as well as non-uniform refined case is compared. Table 5 shows the value of reattachment length for uniform (3600 grid) refined grid is 0.26 m from the step and the value of reattachment length for non-uniform (2100 grid) refined grid is 0.25 m from the step and. The value of reattachment length in case of higher non-uniform (7140 grids) refined grid is 0.25 m from the step. This indicates that there is no significant change in accuracy of the result beyond 2100 non-uniform grids.

Results and discussions

Effect of transitions on reattachment lengths

In the present study, an attempt is made to demonstrate the velocity profiles in the separation region and the variation of reattachment lengths with Reynolds number. Reynolds number, used in the present study is defined as uD/n, where u is the inlet velocity, D is the hydraulic diameter and is equal to twice the inlet height (Assuming the height too small in comparison to the width of the duct, also assumed by Armaly et al. [ 2]) and νis the kinematic viscosity. Armaly et al. [ 2] reported the existence of three recirculation regions, two at the bottom wall and one at the top wall downstream of the step. However, in the present study, flow over a two dimensional backward facing step is studied and only two recirculation regions, one each at the bottom and top walls downstream of the step, are observed, which is in line with the observations of Barton [ 4], Lee and Mateescu [ 5] and Biswas et al. [ 23]. The reattachment lengths for primary separation region at the bottom wall as well as secondary region at the top wall are determined. It is found that these reattachment lengths are function of the Reynolds number.

Figure 3 shows the comparison of the numerical results of variation of primary reattachment length x₁ with Reynolds number R_e with the experimental results of Armaly et al. [ 2]. The laminar regime of the flow can be characterized by the reattachment length that increases with the increase in Reynolds number. However, the increase is nonlinear in nature. The laminar regime for the present study (R_e<900) as identified from Fig. 3 varies with that of Armaly et al. [ 2], (R_e<1200). Whereas, the transitional flow regime can be characterized by the decreasing reattachment length with increase in the Reynolds number. In the transitional regime, the reattachment length first sharply decreases up to the Reynolds number of 2000, then an irregular decrease up to the Reynolds number of 5600 and finally becoming constant around 6000. Armaly et al. [ 2] reported the transitional and turbulent regime to lie in the range of 1200<R_e<6600 and R_e>6600 respectively. Figure 3 reveals that there is agreement between the experimental and numerical values up to R_e = 200 and thereafter a substantial discrepancy is visible. This discrepancy was also reported by Armaly et al. [ 2] and Biswas et al. [ 23]. The difference in the variation of the primary reattachment length x₁ as evident from Fig. 3 is probably due to the limitation of two-dimensional flow in the computations. As also reported in Biswas et al. [ 23], this difference exists even after running the simulations with finer grids on a long computational domain.

Figure 4 shows the variation of the length of the separation point at the upper wall from the step x₂ with the Reynolds number R_e and Fig. 5 shows the variation of the length of the reattachment point from the step x₃ with the Reynolds number R_e. It is evident from these Figs that the secondary recirculation zone exists only for laminar and transitional flow, which was also reported by Armaly et al. [ 2] and Biswas et al. [ 23]. The existence of the adverse pressure gradient at the edge of the step induces the formation of the secondary recirculation region at the upper wall. In both experiments and numerical simulation, the secondary separated region exists after the Reynolds number 400 and it disappears above the Reynolds number around 7000 but the discrepancy in the values of x₂ and x₃ are visible from the Figs. 4 and 5. Also, the Figs. 3–5 show that the secondary recirculation zone at upper wall starts before the reattachment point at the bottom wall and ends after the reattachment point of the primary recirculation region.

Figure 6 shows the variation of reattachment length for primary recirculation region x₁ with the Reynolds number for different expansion ratios. Six different expansion ratios such as 1:1.24, 1:1.38, 1:1.47, 1:1.53, 1:1.94, 1:2.20 are considered for the study. It is found that the reattachment length increases with increase in expansion ratio up to R_e = 800, where the flow is predominantly considered to be two dimensional. Biswas et al. [ 23] reported that even after R_e = 400, the two dimensional flow becomes unsteady in nature showing low frequency oscillations. However, the nature of the variation of x₁ with respect to R_e was not discussed. In the present study, beyond R_e = 800, the variations are chaotic but the reattachment length finally reaches a constant value close to the end of the transition regime. Almost similar trends for the variation of the separation point and the reattachment point at the upper wall are evident from Figs. 7 and 8 respectively. For most of the expansion ratios, the secondary recirculation region ceases to exist beyond R_e = 4800.

Figure 9 shows the variation of detachment and reattachment length with Reynolds number for the upper and bottom walls for the flow over an inclined step of expansion ratio 1:1.94. The step is inclined at 44.5° to the horizontal. It is evident from the Fig. 9 that the trends of the curves are similar to the vertical step but the secondary recirculation zone for the inclined step exists only in the laminar range of 800<R_e<1600. No secondary recirculation zone in transition or turbulent flow is observed at the upper and the bottom walls. Furthermore comparison of the variation of the primary recirculation region for the vertical, inclined and the round edge steps is made in Fig. 10. The primary reattachment length is maximum for the vertical step followed by the inclined and the round step up to R_e = 1200. But beyond that, the reattachment length decreases rapidly for the vertical and the inclined step, whereas it becomes uniform for the round edge step.

Effect of transitions on flow field

The two-dimensional velocity components in x and y are represented by u and v, respectively. Detailed measurements of the vertical distributions of velocity components u and v are plotted in the xy-plane at different longitudinal distances from the step. Figure 11 presents the vertical variations of the longitudinal velocity component u at different distances from the vertical backward facing step at R_e = 1000. It is evident from the profiles that the flow is separated at the step, resulting in the formation of the two recirculation regions downstream of the step, one at the bottom wall and another at the top wall of the duct. The velocity gradient in the y direction is very steep, which weakened toward the reattachment point. Also, significant negative velocities are observed close to the near wall region at the bottom wall downstream of the step. Afterwards, the flow redeveloped to a fully developed parabolic profile across the whole duct. Although, similar trends are observed for the inclined and the round edged step, yet there are differences in the reattachment lengths at the top and the bottom walls as indicated in Figs. 9 and 10. Figure 11 also demonstrates that the recirculation region at the upper wall starts earlier than the reattachment point at the bottom wall as reported by Armaly et al. [ 2]. Similar observations are also made for the inclined and the round edged transitions. Figure 12 represents the vertical variations of the vertical velocity component v at different longitudinal distances from the vertical backward facing step at R_e = 1000. Strong downward motion is noticed in the primary recirculation zone at the bottom wall downstream of the step, whereas, adjacent to the step, feeble upward movement is noticed. This signifies the presence of a strong primary recirculation region adjacent to the step. The evidence of the recirculation regions at the upper and the bottom walls are much clear from the plot (Fig. 13) of the longitudinal velocity along the duct at a distance of 0.6 mm from the bottom and the upper walls. The variation of the longitudinal velocity close to the bottom wall is similar for all the transitions. The maximum negative velocity for all the transitions occurs at approximately a distance of 0.26 m from the step. Only for the vertical transition, the magnitude of the maximum negative velocity is higher in comparison to the inclined and the round edged steps. Also, it is evident from Fig. 13 that the primary reattachment length at the bottom wall is minimum for the inclined transition. The trends for the secondary recirculation zones at the top wall are same for the vertical and round edged steps, the recirculation region being the minimum for the vertical step among all the transitions. Figure 14 presents the comparison of the vertical variations of the longitudinal velocity component u at different longitudinal distances from the backward facing steps at turbulent regime (R_e = 10000). Although primary recirculation region at the bottom wall is observed for vertical and the round edged transitions, but it is much shorter for the round edged backward facing step at the turbulent regime. Existence of fully developed turbulent flow at a distance of 0.2 m from the step is evident for the inclined transition. The primary recirculation regions are also evident from the contour plot of the longitudinal velocity component downstream of the vertical step across the whole computational domain in Fig. 15, which also agrees well with the results of Lee and Mateescu [ 5]. To check the effect of different turbulence models on the accuracy of result, the simulated results are compared using various turbulence models. The turbulence models chosen in this study for comparison are standard k-ϵ model, standard k-w model, spalart allamaras model and Reynolds stress model. Figure 16 presents the comparison of vertical variations of the longitudinal velocity component u at different distances from the step at R_e = 10000 using various turbulence models. Similarly Fig. 17 presents the comparison of vertical variations of the vertical velocity component v at different distances from the step at R_e = 10000 using different turbulence models. From Fig. 16 and Fig. 17, it is concluded that the use the turbulence models for the simulation of flow over backward facing step in the present study have negligible effect on the accuracy of the result.

The streamwise variations of longitudinal wall shear stress at the bottom and the upper walls downstream of all the three transitions are presented in Fig. 18. The bottom wall profiles show a dominant negative peak corresponding to the reverse flow in the primary recirculation zones downstream of the steps, the magnitude for the round edged transition being the minimum. The negative peak is followed by a sharp recovery and finally a constant value for the fully developed turbulent flow [ 22]. The wall shear stress variations at the upper wall show similar trends of the variation of the longitudinal velocity component close to the upper wall, as presented in Fig. 13.

Conclusions

The present study investigated the study of the velocity distribution and the separation phenomenon of flow of air over a two dimensional backward facing step. The variation of reattachment lengths for all three cases are analyzed for wide range of Reynolds number ranging from 100 to 7000 covering the laminar, transition and turbulent flow of air. Flow over steps with various expansion ratios of 1:1.24, 1:1.38, 1:1.47, 1:1.53, 1:1.94, 1:2.20 are also carried out to study the effect of different expansion ratios on the reattachment length. The primary reattachment length is maximum for the vertical step followed by the inclined and the round step up to R_e = 1200 and beyond that, the reattachment length decreases rapidly for the vertical and the inclined step, whereas it becomes uniform for the round edge step. The primary reattachment length increases with increase in the expansion ratios of backward facing step. The bottom wall shear stress profiles for the three transitions show a significant negative peak corresponding to the reverse flow in the primary recirculation zone downstream of the step. The magnitude of the peak is minimum for the round edged transition. Also the effect of different turbulence models on the accuracy of result are compared using various turbulence models in the simulation and it is found that the use the different turbulence models for the simulation of flow over backward facing step in the present study have negligible effect on the accuracy of the result. The understanding of flow characteristics in the separated region will give an idea to design the structure in an alternate way so as to achieve the desirable characteristics such as reduced drag, enhanced mixing, controlling of noise and vibration etc.

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