1 Introduction

Unstable separated flows have received particular attention due to their presence in many practical applications. Among them, the flow in backward-facing steps, which is representative of several industrial geometries such as weirs, combustion chamber and the sudden enlargement, causes the formation of numerous vortex structures that can play an important role in the phenomena of mixing or combustion [1,2,3,4,5,6,7].

The hydrodynamic characteristics of Newtonian fluids have been studied by many researchers [8,9,10,11]. Among others, Louda et al. [12] investigated the effect of 3D for the stationary conditions and they reported that these effects have a little importance. The flow through this geometrical configuration is known by its detachment behind the step [13]. The rotating flow backward the step increases the instabilities and interaction between the hot and cold mixed particles of fluid [14]. The combination of the hydrodynamic and thermal instabilities plays an important role on the friction factor (Cf) and Nusselt number (Nu) [14].

Other researchers examined the influence of inclination angle of step [15]. Choi et al. [16] investigated the effect of seven geometrical configurations (10°, 15°, 20°, 25°, 30°, 45° and 90°) for different Reynolds numbers (5000, 8000, 11,000, 15,000, 47,000 and 64,000). They found that the reattachment length remains almost constant for Re > 15,000, even when the inclination angle is increased further than 30°.

With the help of the computational method and for a range of Reynolds number of 36-4500, Macagno and Hung [17] studied the axisymmetric flows of Newtonian fluids through sudden enlargements with an aspect ratio of 2. They reported that the primary task of the eddy for the laminar condition is to shape the flow with a quite low-energy exchange. For a Newtonian fluid flowing through a 1:3 planar symmetric expansion under laminar conditions, Durst et al. [18] observed by experiments two symmetric vortices across the walls of expansion at Re = 56. They observed also the flow bifurcation at Re = 114 with vortices of disparate sizes in both of the prominent corners. For Reynolds number in the range 70–8000, Armaly et al. [9] determined by experiments the velocity distribution and reattachment lengths of a Newtonian fluid through a single backward-facing step. Oliveira and Pinho [19] also explored the flow characteristics of a Newtonian fluid within a backward-facing step, but under laminar flow conditions. For Newtonian fluids, the increase in the expansion ratio may decrease the critical Reynolds number for the symmetry-breaking bifurcation, as reported by Drikakis [20]. At Reynolds number equal to 40.5 and for a 1:3 an axisymmetric abrupt enlargement, Fearn et al. [21] recorded a similar flow bifurcation. For a range of Re from 20 to 211, Hammad et al. [22] studied experimentally the reattachment and redevelopment lengths of a Newtonian fluid flowing through a planar enlargement with an area ratio of 2.

In fact, the fluids flowing via flow devices and through various real-world scenarios are rheologically complex and have a non-Newtonian behavior. Some works have been achieved on this kind of fluids through channels with abrupt enlargement. For the laminar regime and shear-thinning fluids, Mishra and Jayaraman [23] explored by experiments the asymmetric flow patterns through a planar abrupt enlargement with a ratio ER = 16. For power-law fluids and low Re (from 30 to 125), Manica and De Bortoli [24] studied the flow through a 1:3 planar abrupt expansion. They reported that the bifurcation flow for Newtonian fluids occurs at a critical Reynolds number lower than that for the shear-thinning fluids, whereas the shear-thickening fluid shows the least values of critical Re. For low Reynolds number and power-law fluids with a range of power-law indices from 0.3 to 3, Neofytou [25] used the Casson and power-law models to assess the transformation of symmetric to an asymmetric flow within 1:2 sudden expansion.

Ternik et al. [26] used two models, namely the quadratic viscosity and power-law models, to study the flow of shear-thickening fluids at low Re; they observed a significant effect of the shear-thickening behavior on the flow asymmetry. In another work, Ternik [27] interested in the power-law fluids (with power-law indices from 0.6 to 1.4) at Reynolds numbers from 10−4 to 10. They observed a small recirculation, common of creeping flow for the shear-thinning fluids where decreasing the size and strength of the secondary flow. Ameur [28] studied the 3D flows of shear-thinning fluids through sudden expending pipes. He explored the effect of power-law index n (from 0.4 to 1), Reynolds number Re (from 0.1 to 600), spacing between the branching channels, number of branching channels and the expansion ratio. He observed a considerable increase in the pressure losses with the increase in Re and n and decreased expansion ratio. However, he found only a little increase in Δp with decreased spacing ratio, while Δp remained almost the same with increased number of branching channels.

Our search in the literature revealed that the flow of shear-thinning fluids through backward-facing steps for a range of Reynolds number covering the laminar, transitional and turbulent flow regime has not been studied previously. Therefore, the purpose of this paper is to explore the flow patterns of this kind of fluids through a wide range of Re. The Reynolds number is varied between 100 and 12,000.

2 Presentation of the problem

Geometry of the problem studied is a backward-facing step (Fig. 1). Lengths of the ducts before and after the step are L1 = 200 cm and L2 = 100 cm, respectively. These values are sufficient to obtain a developed flow at the entrance region of the duct and to avoid reverse flow at the exit section. Effect of the expansion ratio (ER) is examined by realizing five geometries (ER = S/H = 0.250, 0.375, 0.500, 0.625 and 0.750), where the duct width is taken as W = 20 cm. We note here that the side walls of the duct are considered as symmetric, i.e., the problem is treated as two-dimensional, and thus, this value of W has no effect of the flow patterns.

Fig. 1
figure 1

Problem studied

Full size image

The working fluid is CMC (carboxymethylcellulose) with the following rheological properties: concentration c = 0.04 [g CMC/g], consistency index m = 0.79 [Pa sn] and power-law index n = 0.83. This fluid has a shear-thinning behavior modeled by the Ostwald model.

3 Theoretical background

The equations of continuity and momentum for the case studied are given as follows:

$$\frac{\partial }{{\partial x_{i} }}\left( {\rho u_{i} } \right) = 0$$
(1)
$$\rho \frac{{{\text{d}}u_{i} }}{{{\text{d}}t}} = \rho g - \nabla p_{i} + \mu_{a} \nabla^{2} u_{i}$$
(2)

where μa is the apparent viscosity of the non-Newtonian fluid. For a shear-thinning fluid, the Reynolds number is defined as:

$$\text{Re} = \frac{{\rho u_{0}^{2 - n} D_{h}^{n} }}{m}\left( {\frac{3n + 1}{4n}} \right)^{ - n} 8^{1 - n}$$
(3)

where D is the hydraulic diameter, u0 is the inlet velocity and ρ is the fluid density.

The friction coefficient on the wall permits to determine the point of reattachment flow. It is defined as:

$$C_{f} \left( x \right) = \frac{2}{\text{Re}}\left( {\frac{\partial u}{\partial y}} \right)_{\text{wall}}$$
(4)

In dimensionless form, we define the following parameters:

$$X^{*} = \frac{x}{H};\;Y^{*} = \frac{y}{H};\;Z^{*} = \frac{z}{H};\;S^{*} = \frac{S}{H}$$
(5)

4 Numerical details

The geometry of the computational domain has been created by the CFD software ICEM (Ansys Inc.). Calculations have been achieved with the help of computer tool CFX (version 18.0) which is based on the finite volume method to solve the governing equations. The Reynolds number is varying in a wide range covering the laminar, transitional and turbulent flow regimes. The numerical study is achieved for two-dimensional steady-state, isothermal and incompressible fluid. The second-order upwind scheme was used for the convective terms and the SIMPLEC (Semi-Implicit Method for Pressure-Linked Equations-Consistent) algorithm for the pressure–velocity coupling. The standard k − ε model is used for the turbulent flow regime. The Boussinesq equation is applied for the flow, with the following boundary conditions: A parabolic profile of velocity is set at the inlet section (Poiseuille), and a Neumann condition is set at the outlet section of duct. At the walls, the no slip boundary condition is set.

Hexahedral grid elements (Fig. 2) were used to discretize the computational domain, and the increased mesh density was created in the downstream region of channel in order to determine the details of the fluid boundary layer. Mesh independency was checked by comparing the converged results of the reattachment length (Table 1). The size function of mesh elements was kept constant during the refinement process, while the size of elements near the walls was reduced. With the mesh of about 0.8 elements, the obtained results of the reattachment length (Xu) did not change by more than 2%. Therefore, this was adopted for further simulations.

Fig. 2
figure 2

Computational domain meshed with hexahedral elements

Full size image
Table 1 Mesh independency tests at Re = 7 × 103
Full size table

With a machine (INTEL® i7 processor with 8 Gb RAM) and for a residual target 10−6, the convergence was achieved after about 3500-4000 iterations, which corresponds to about 25–30 h. CPU time.

5 Results and discussion

5.1 Validation

The accuracy of the predicted results and the correctness of the numerical method have been checked in this section. The experimental works of Hu et al. [29] for a Newtonian fluid and that of Kahine et al. [30] for a shear-thinning fluid have been taken as references for the validation of our numerical results. With the same geometrical configuration and the same fluids as those used by these authors, the results of pressure coefficient and dimensionless axial velocity are presented in Fig. 3a, b, respectively. Furthermore, a comparison between the k-ε and the SST models against an experimental data of Hu et al. [29] has been made, as illustrated in Fig. 3a. The comparison between both models and the experimental data of Hu et al. [29] as well as that of Kahine et al. [30] reveals a good agreement. Other researches [31] have used the k − ε model for the flow through backward-facing steps and they obtained satisfactory results.

Fig. 3
figure 3

Validation of our predicted results, a pressure coefficient versus dimensionless channel length, b axial velocity for n = 0.6, d/D = 0.112

Full size image

5.2 Flow patterns

The backward-facing step (BFS) is a simple and common case to explore the flow behavior. It has been used as a test case to explore the validity and efficiency of numerical methods. In the present geometrical configuration, the fluid is subjected to a sharp expansion where the main flow is separated. A vortex is formed behind the step, where strong adverse pressure gradients and a strong interaction of the different flow structures with the solid wall are present. The dividing streamline from the step tip to the reattachment location Xu defines the shear layer. The line that separates between the primary and secondary recirculation loops is the line that is located between the reattachment and the reseparation points and it is known as the primary–secondary interface line.

Results on the effect of Reynolds number are presented in this section. The streamlines depicted on Fig. 4 illustrate the vortex structure developed behind the step. The important observation in this figure is the presence of a secondary recirculation zone at the top wall of the duct, from Re = 400 (Fig. 4b). With increased Re, the flow on the central axis is intensified and the secondary recirculation zone on the upper wall disappears (Fig. 4c).

Fig. 4
figure 4

Streamlines for different Reynolds numbers, a Re = 100, b Re = 400, c Re = 3000

Full size image

For low values of Reynolds number, the flow remained symmetric with a separation location in close proximity of the enlargement corner, the overall size of which is improved with the raise of Re. The increased Re turns the flow into asymmetric, with different separation areas of varying lengths that can be linked to the lower walls of the corresponding conduit.

The critical state is a stability exchange point for the symmetric and axisymmetric conditions. When the value of Reynolds number is sufficiently low, the flow is unable to sustain the disturbance that diminishes via viscous dissipation. The viscous dissipation decreases with increased Re, and the symmetric flow will be altered to stability, as mentioned by Hawa and Ruzak [4].

Even though the BFS is a simple case, it has some fundamental problems that could be transferred into much complicated cases. It is considered to be an optimal geometry for separated flows, due to the wake dynamics in the downstream region of the step.

5.3 Reattachment length

Our predicted results of the reattachment length vs. Reynolds number have been compared against the experimental data of Armaly et al. [9] and Nie et al. [32] and presented in Fig. 4. We note that the same fluid, as well as the same geometrical configuration as those used by these authors, has been taken in consideration, as detailed bellow. Values of the length (L1), width (W) and height (h) in the upstream portion of the channel in this section are as follows: 200 cm, 8.0 cm and 0.98 cm, respectively. The values of length (L2), width (W) and height (h) in the upstream portion of the channel are as follows: 100 cm, 8.0 cm and 1.98 cm, respectively. This geometrical configuration gives a BFS of 1.0 cm in height, an expansion ratio of 2.02:1 and an aspect ratio of 8 at the step. Air has been considered here as a working fluid. As illustrated in Fig. 4, the comparison between our results and those of Armaly et al. [9] and Nie et al. [32] shows a satisfactory agreement.

At the level of expansion, a shear layer generated by the gradient of velocities develops spatially. It interacts with the primary recirculation zone that forms at the bottom wall. This recirculation zone is characterized by an important parameter named reattachment length. Beyond the point of reattachment, the flow tends to develop with a negative pressure gradient. This part of the wall is called the recovery zone. Figure 5 shows the distribution of the reattachment length for the primary recirculation zone as a function of Reynolds number. We note that the numerical values obtained with our computer code show a good agreement with the experimental values of other authors. As the Reynolds number continues to increase, the rate of amplification of instabilities also increases and the intensity of vortex increases. This results in a strong modification of the recirculation zone due to the efficient convective transport of fluid particles to the duct wall. Figure 5 shows that the recirculation length continues to increase up to Re = 1000. For greater Reynolds numbers, the recirculation length decreases, and then, it stabilizes toward Re = 3000 at a value between 5 and 6H. (We remind that Xu = recirculation length/H.)

Fig. 5
figure 5

Reattachment length for different Reynolds numbers

Full size image

For the laminar flow conditions (Re between 20 and 211), Hammad et al. [22] observed that the reattachment and redevelopment lengths were generally a linear function of Re. However, the recirculation eddy strength was nonlinear and dependent on Re. They reported also that the recirculation eddy strength becomes weak with the raise of Re.

The separation location in the downstream region behind the step is a single fixed point where the flow separation occurs. The BFS illustrates all important simplifications of several industrial problems with respect to more complicated configurations.

5.4 Axial velocity

Figure 6 describes the flow patterns along the duct. The upstream boundary layer takes off at the top of the step to give rise of the shear layer due to the presence of a strong adverse pressure gradient. The line of separation between the mean flow and the recirculation zone is slightly curved, and the shear layer is thin enough not to be affected by the presence of the wall. In this region, a part of the flow is deflected upstream to generate a recirculation zone, while the other part is redirected downstream with a decreased size of swirling flows.

Fig. 6
figure 6

Axial velocity for Re = 8000, S* = 0.5, Z* = 1

Full size image

These structures are intense and visible, especially near the step, and they will be weak further away from the expansion section (Fig. 6).

The expansion ratio (RE) is a measure of the adverse pressure gradient acting on the shear layer as well as a measure of the favorable pressure gradient acting on the recirculation region. A low RE minimizes the effects of the pressure gradient due to expansion (Fig. 7).

Fig. 7
figure 7

Axial velocity for Re = 8000, X* = 1, Z* = 1

Full size image

5.5 Turbulent kinetic energy

Figure 8 shows the distribution of the turbulent kinetic energy (K* = K/ρu2). This energy is defined by the expression K = 0.5*((u’2) + (v’2) + (w’2)). It can be seen that the maximum values of turbulent kinetic energy are reached near the step. This is explained by the presence of higher interaction between the fluid particles just behind the step. Also, the turbulent kinetic energy increases with decreased expansion ratio (Fig. 9).

Fig. 8
figure 8

Turbulent kinetic energy for Re = 12000, S* = 0.5, Z* = 1

Full size image
Fig. 9
figure 9

Turbulent kinetic energy for Re = 12000, Y* = 0.025, Z* = 1

Full size image

6 Conclusion

The flow through a backward-facing step (BFS) is characterized by a detachment of the fluid on a sharp edge at 90° and by the formation of vortices downstream of the step. The point of reattachment is the equilibrium position between the vortex and the main flow.

The fluid entrained yields an adverse pressure gradient, thus resulting in the reattachment of the shear layer. The reattachment length, which is considered as one of the primary parameters of the wake characteristics of the BFS flow, is found to be increased in the laminar flow regime and it remained almost constant at the turbulent regime, from Re = 5000.

The vortex size increases and moves downstream as Reynolds number increases in the laminar regime, it decreases and moves upstream for the transient regime, and it decreases or remains almost constant for the turbulent flow. The obtained results show the formation of a secondary vortex at the upper wall for Re = 400, which disappears with increased Re. Finally, the increase in expansion ratio yields a decrease in the turbulent kinetic energy.