1 Introduction

Understanding the behavior of gas–solid turbulent flow in sudden expansions is critical for designing and optimizing many industrial processes such as fluidized bed reactors, pneumatic conveying systems, and cyclone separators. These are just a few examples of the many industrial processes where understanding gas–solid turbulent flow in sudden expansions is important. Researchers have developed a range of experimental and numerical techniques to study these flows and improve our understanding of the underlying phenomena. By improving our understanding of these flows, we can optimize these processes for improved efficiency, reliability, and product quality [1,2,3,4,5,6].

In gas–solid flow, the sudden expansion can cause increasing a range of complex flow phenomena, turbulence, particle–particle and particle–wall interactions, and changes in particle trajectories, all of which can have significant effects on the flow behavior. Gas–solid flows can exhibit different flow regimes, depending on the gas velocity, particle concentration, and particle characteristics [7, 8]. During sudden expansions, changes in these parameters can lead to transitions between flow regimes. For example, a transition from a dispersed flow regime, where the particles are uniformly distributed in the gas phase, to a cluster-dominated regime can occur due to the expansion-induced changes in the flow dynamics. The transition can affect the particle transport, mixing, and overall flow behavior [9]. The interaction between the gas and solid phases can result in complicated flow patterns and variations in the pressure distribution in a gas–solid turbulent flow through a sudden expansion. This pressure difference occurs due to the conversion of kinetic energy into potential energy as the flow expands. The pressure can affect the overall flow behavior, particle motion, and deposition patterns. In particle dispersion, the sudden expansion can cause the dispersion of solid particles within the gas phase. As the flow expands, the particles experience different forces and velocities, leading to the dispersion of particles in various directions. This dispersion can affect the particle distribution and deposition patterns downstream of the expansion [10,11,12,13,14].

The Eulerian and Lagrangian approaches are two different mathematical frameworks used to describe and analyze the motion of solid particles in two-phase flow (gas–solid flow) [15,16,17,18,19]. The Eulerian approach focuses on describing the fluid flow properties at fixed points in space. In this framework, the fluid flow is characterized by variables such as velocity and pressure, which are defined at each point in the fluid domain. The Eulerian approach is well-suited for analyzing large-scale fluid flow phenomena, such as weather patterns or ocean currents, where the motion of individual solid particles is less important than the overall behavior of the fluid [20,21,22]. The Lagrangian approach, on the other hand, follows the motion of individual solid particles over time. In this framework, each solid particle is tracked as it moves through the flow field, and its properties, such as position, velocity, and acceleration, are described as functions of time [23,24,25,26]. The choice of approach depends on the goals of the analysis and the level of detail required to understand the system being studied. In many cases, a combination of both approaches may be necessary to gain a comprehensive understanding of the fluid dynamics.

The review of Kumar et al. [27] sheds light on studies single-phase flow of sudden expansion. A critical test that can be used to evaluate different turbulence models is also thought to be a part of this flow. The flow over a sudden expansion is commonly used as a case study for testing for several CFD codes and models relating to turbulent flow. Several turbulence models have been utilized, including RANS, LES, and DNS, to simulate this flow in the 2D and 3D domains [28,29,30,31]. However, many researchers have extensively studied the turbulent single-phase flow through sudden expansion and backward-facing steps [32,33,34,35,36].

The performance of gas–solid two-phase flow was examined experimentally by Arefmanesh and Michaelide [37] using the pressure recovery phenomenon over a fairly constrained range of operational parameters. For different flow orientations, another group of experiments links the extra loss coefficient to the flux Richardson number. Richardson number relates the ratio of buoyancy forces to the shear forces present in the flow. Additionally, these investigations [38,39,40,41] come to the conclusion that the link is only valid below a particular area ratio. In these systems, particle dispersion has been linked to a number of causes, one of which is turbulence [42].The development and assessment of turbulence model performance were the subject of a different collection of studies [43,44,45,46,47]). These studies seek to improve particle–turbulence interaction modeling. In terms of the loss coefficients, the results of abrupt and progressive expansions have also been stated. These literatures have examined the loss coefficients in terms of a number of operating parameters for various flow orientations [48].The purpose of the experiment [49] was to show how the wide-angled diffuser behaves at various Reynolds numbers and angles. This experiment demonstrated how velocity and pressure recovery were impacted by the Reynolds number and angles (30°, 90°). When the velocity doubled from 10 to 20 m/s, this experiment showed a substantial difference; the increase in wall static pressure recovery increased by 8.31%. The Reynolds number was increased to enhance pressure recovery.

Many reviews had beaning done to turbulent gas–solid flow to shed light to many factors. This review [50] article focused specifically on gas–solid flow in pipes, which is an important area of study for applications such as pneumatic conveying and cyclone separators. The authors discuss the various flow regimes that can occur in vertical pipes, as well as the factors that influence particle transport and deposition. Also, [51, 52] discussed the various hydrodynamic models that have been developed to describe gas–solid flow, including the two-fluid model, the kinetic theory approach, and the discrete particle model. The authors also provide examples of how these models have been applied to specific industrial processes, such as fluidized bed reactors and pneumatic conveying systems.

The gas–solid phase was simulated in an Eulerian–Eulerian frame using Reynolds-averaged Navier–Stokes equations (RANS) with the software ANSYS Fluent. That helped to study the axial velocity, pressure recovery, volume fraction, and kinetic energy. In comparison with alternative combinations of turbulence models such as: standard k-\({\varvec{\varepsilon}}\) model,RNG + SWF and Realizable k + SWF, and wall functions, the calculated results using the standard k-\({\varvec{\varepsilon}}\) model and standard wall function were shown to produce results that are most similar to experimental data of [44]. To predict their effects on two-phase pressure recovery, the specularity coefficient, and the coefficient of restitution for particle–particle collisions, two important numerical parameters that consider particle–wall collision and particle–particle collision, respectively, were each taken into consideration separately. Because the recirculating zone absorbs the lost energy, either a decrease in particle–particle collisions or an increase in particle–wall collisions results in a lower pressure recovery. The conclusion was that lighter particle movement allows for higher pressure recovery. In a similar vein, it was discovered that pressure recovery rose as inlet slip velocity ratio increased [22]. After that [53] completing their work for different area ratios, which showed two counter-acting effects on the pressure recovery phenomena. The volume loading ratio had been taken between 2 and 10%. Collisions were found to play a significant influence and the collisions affected the pressure recovery and reattachment length. Two conflicting effects on the pressure recovery are visible in the area ratio. The pressure recovery rises up to a critical area ratio value (9.97) and falls down after that. However, an increase in the area ratio always results in an increase in the reattachment length. The pressure recovery and reattachment length decrease with increasing particle size and solid phase density. Therefore, with a sudden expansion, the gas phase receives the energy from the lighter and smaller particles more than the larger particles.

El-Askary et al. [23] discovered that the coefficient of loss in the presence of tiny particles reduces in value with increasing mass loading ratio. This discovery resulted from their Eulerian–Lagrangian analysis of the performance of a pipe with a sudden expansion. At various particle sizes, mass loading ratios, and Reynolds numbers, the performance of abrupt expansion in the form of loss coefficient is investigated. The performance of the abrupt expansion can be enhanced by the smallest particles, according to the computational findings, while the size of the separation bubble grows. Furthermore, even at very low solid loadings, the performance of the numerical model is significantly enhanced by the inclusion of inter-particle [8, 45].For gas–solid flows, Tashiro and Tomita [38] investigated the loss coefficient for the vertical downward and upward sudden expansions, respectively, while Deguchi et al. [39] analyzed the loss coefficient for the horizontal sudden expansion.

Completing the study of the single-phase flow [54], which was simulated with using the finite volume method and the Reynolds-averaged Navier–Stokes equations (RANS). By the house developed FORTRAN code to simulate this flow, the elements for the turbulent flow with the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm for pressure–velocity coupling. An Eulerian–Eulerian method is used in this work to understand the mechanisms driving turbulent flow during abrupt expansion. By comparing the experimental result, the numerical code has been validated. In this research, the critical variables will be the area ratios, particle diameter, and void fraction, which examining on the local skin fraction, pressure, velocity (gas–solid), solid concentration, streamlines, and turbulent kinetic energy behind the step wall.

2 Mathematical model

2.1 Problem definition

Figure 1 displays the schematic representations of the two-dimensional abrupt expansion. This sketch illustrates the diffuser at a 90° angle.

Fig. 1
figure 1

Sudden expansion geometry

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2.2 Governing equation

The governing equation for gas–solid turbulent flow in sudden expansion can be described by the conservation equations for mass, momentum, and energy. The flow is the steady flow and incompressible. The continuity equation represents the conservation of mass, accounting for the inflow and outflow rates of both the gas and solid phases. The momentum equation considers the forces acting on the gas and solid particles, including pressure gradients, viscous forces, and turbulent stresses. It accounts for the interaction between the phases and their respective velocities. Additionally, models for turbulence, particle–particle interactions, and particle–wall interactions are incorporated to capture the complexities of the flow. Solving these equations numerically provides insights into the behavior of gas–solid turbulent flow in sudden expansion and aids in test of validity of model and performance [22, 53]. The flow through the sudden expansion consideration in this study. The Eulerian–Eulerian approach is used to code the numerical model, by solving Reynolds average Naiver–Stokes equations (RANS) and the two-equation k-ε turbulence model.

2.2.1 Gas phase

Continuity equation for the gas phase as shown in Eq. (1).

$$ \frac{\partial }{\partial t}\left( {\alpha_{{\text{g}}} \rho_{{\text{g}}} } \right) + \frac{\partial }{{\partial x_{{\text{j}}} }}\left( {\alpha_{{\text{g}}} \rho_{{\text{g}}} U_{{{\text{g}},{\text{j}}}} } \right) = 0 $$
(1)

where \(\alpha \) is volume fraction, \(\rho \) is density, \(U\) is the velocity and subscript (g for gas and s for solid).

Momentum equation for the gas phase as shown in Eq. (2).

$$ \begin{array}{*{20}c} {\frac{\partial }{\partial t}\left( {\alpha_{{\text{g}}} \rho_{{\text{g}}} U_{{\text{g}}} } \right) + \frac{\partial }{{\partial x_{j} }}\left( {\alpha_{{\text{g}}} \rho_{{\text{g}}} U_{{\text{g,j}}} U_{{\text{g,i}}} } \right) = - \frac{{\partial \left( {\alpha_{{\text{g}}} P} \right)}}{{\partial x_{{\text{i}}} }}} \\ { + \frac{\partial }{{\partial x_{{\text{j}}} }}\left( {\alpha_{{\text{g}}} \left( {\tau_{{\text{g,ij}}} + \tau_{{\text{g,ij}}}^{\prime } } \right) + \alpha_{{\text{g}}} \rho_{{\text{g}}} g_{{\text{i}}} + F_{{\text{D}}} + F_{{{\text{Lift}}}} } \right.} \\ \end{array} $$
(2)

where \({\tau }_{\text{g},ij}\) is mean gas phase stress tensor,\(\tau_{{{\text{g}},ij}}^{\prime }\) is Reynolds stress,\({g}_{i}\) is a gravity, \({F}_{\text{D}}\) is a drag force, and \({F}_{\text{Lift}}\) is a lift force.

The kinetic energy of turbulence's transport as shown in Eq. (3)

$$ \begin{array}{*{20}c} {\frac{\partial }{\partial t}\left( {\alpha_{g} \rho_{g} k_{g} } \right) + \frac{\partial }{{\partial x_{j} }}\left( {\alpha_{g} \rho_{g} U_{g,j} k_{g} } \right) = \frac{\partial }{{\partial x_{j} }}\left[ {\alpha_{g} \left( {\mu_{g} + \frac{{\mu_{t,g} }}{{\sigma_{k} }}} \right)\frac{{\partial k_{g} }}{{\partial x_{j} }}} \right]} \\ { + \alpha_{g} \tau_{g,ij}{\prime} \frac{{\partial U_{g,i} }}{{\partial x_{j} }} - \alpha_{g} \rho_{g} \varepsilon_{g} + \alpha_{g} \rho_{g} \mathop \prod \limits_{k,g} } \\ \end{array} $$
(3)

and similarly, the transport equation for dissipation rate of turbulence as shown in Eq. (4)

$$\begin{array}{c}\frac{\partial }{\partial t}\left({\alpha }_{g}{\rho }_{g}{\varepsilon }_{g}\right)+\frac{\partial }{\partial {x}_{j}}\left({\alpha }_{g}{\rho }_{g}{U}_{g.j}{\varepsilon }_{g}\right)=\frac{\partial }{\partial {x}_{j}}\left[{\alpha }_{g}\left({\mu }_{g}+\frac{{\mu }_{t,g}}{{\sigma }_{\varepsilon }}\right)\frac{\partial {\varepsilon }_{g}}{\partial {x}_{j}}\right]\\ +{C}_{1\varepsilon }{\alpha }_{g}\frac{{\varepsilon }_{g}}{{k}_{g}}{\tau }_{g,ij}{\prime}\frac{\partial {U}_{g,i}}{\partial {x}_{j}}-{C}_{2\varepsilon }{\alpha }_{g}{\rho }_{g}\frac{{\varepsilon }_{g}^{2}}{{k}_{g}}+{\alpha }_{g}{\rho }_{g}\prod_{\varepsilon ,g} \end{array}$$
(4)

where \({k}_{g}\) \({\varepsilon }_{g}\) are turbulent kinetic energy and turbulent kinetic energy dissipation rate, respectively. Dynamic viscosity of gas (laminar) and turbulent viscosity are this symbol (\({\mu }_{g} , {\mu }_{t,g}\)), respectively,\({C}_{1\varepsilon }\),\({ C}_{2\varepsilon }\), \({\sigma }_{k}\) is the model constants. The additional terms involving \(\prod_{\text{k},\text{g}}\) and \(\prod_{\text{k},\text{g}}\) arise from the presence of the particles and were modeled using the expressions given in Elghobashi and Abou-Arab [55]. The expression for the mean gas phase stress tensor as shown in Eq. (5)

$${\tau }_{g,ij}={\mu }_{g}\left(\frac{\partial {U}_{g,i}}{\partial {x}_{j}}+\frac{\partial {U}_{g,j}}{\partial {x}_{i}}\right)-\frac{2}{3}{\mu }_{g}\frac{\partial {U}_{g,k}}{\partial {x}_{k}}{\delta }_{ij}$$
(5)

where \({\delta }_{ij}\) is Kronecker delta.

The definition of the Reynolds stress tensor as shown in Eq. (6)

$$ \tau_{{{\text{g}},{\text{ij}}}}^{\prime } = - \rho_{{\text{g}}} \overline{{u_{{\text{i}}}^{\prime } u_{{\text{j}}}^{\prime } }} $$
(6)

where \(u_{i}^{\prime }\) is fluctuating velocity.

To which the Boussinesq approximation was used to model as shown in Eq. (7)

$$ \overline{{u_{{\text{i}}}^{\prime } u_{{\text{j}}}^{\prime } }} = - 2\left( {\frac{{\mu_{{\text{t,g}}} }}{{\rho_{{\text{g}}} }}} \right)S_{{\text{g,ij}}} + \frac{2}{3}\rho_{{\text{g}}} \left( {k_{{\text{g}}} + \mu_{{\text{t,g}}} \frac{{\partial U_{{\text{g,m}}} }}{{\partial x_{{\text{m}}} }}} \right)\delta_{{i{\text{j}}}} $$
(7)

2.2.2 Solid phase

Governing equation for solid phase as shown in Eq. (8)

$$\frac{\partial }{\partial t}\left({\alpha }_{\text{s}}{\rho }_{\text{s}}\right)+\frac{\partial }{\partial {x}_{j}}\left({\alpha }_{\text{s}}{\rho }_{\text{s}}{U}_{\text{s},j}\right)=0$$
(8)

Momentum equation for particle phase as shown in Eq. (9)

$$\begin{array}{c}\frac{\partial }{\partial t}\left({\alpha }_{s}{\rho }_{s}{U}_{s}\right)+\frac{\partial }{\partial {x}_{j}}\left({\alpha }_{s}{\rho }_{s}{U}_{s,i}{U}_{s,j}\right)=-\frac{\partial \left({\alpha }_{s}P\right)}{\partial {x}_{i}}-\frac{\partial {P}_{s}}{\partial {x}_{i}}+\frac{\partial {\tau }_{s,ij}}{\partial {x}_{j}}\\ +{\alpha }_{s}{\rho }_{s}{g}_{i}+{F}_{D}+{F}_{Lijt}.\end{array}$$
(9)

where \(P\) is a pressure of gas, \({P}_{s}\) is a pressure solid recovery.

Modeling of the Stresses in the Particle Phase. A solid phase tension and solid pressure are created when the particles collide. To simulate these contributions, the kinetic theory of granular flow (KTGF) suggested by Ding and Gidaspow [56] is used in the current work. Solid phase stress is expressed as a tensor as shown in Eq. (10)

$$\begin{array}{c}{\tau }_{s,ij}=-{p}_{s}{\delta }_{ij}+2{\alpha }_{s}{\mu }_{s}{S}_{s,ij}+{\alpha }_{s}\left({\lambda }_{s}-\frac{2}{3}{\mu }_{s}\right){S}_{s,kk}{\delta }_{ij}\\ {S}_{s,ij}=0.5\left(\frac{\partial {U}_{si}}{\partial {x}_{j}}+\frac{\partial {U}_{sj}}{\partial {x}_{i}}\right).\end{array}$$
(10)

where \({S}_{s,ij}\) is mean strain-rate tensor.

The solid pressure is given as [57] as shown in Eq. (11):

$${P}_{s}={\alpha }_{s}{\rho }_{s}{T}_{s}+2{\rho }_{s}\left(1+e\right){\alpha }_{s}^{2}{g}_{0,ss}{T}_{s}$$
(11)

where \(e\) is \(\text{Coefficient of restitution for inter particle collision}\) and \({T}_{s}\) is \(\text{Granular temperature}\).

The radial distribution function is [57] as shown in Eq. (12)

$${g}_{0}={\left(1-{\left(\frac{{\alpha }_{s}}{{\alpha }_{s,max}}\right)}^\frac{1}{3}\right)}^{-1}$$
(12)

where \({\alpha }_{s,max}\) is the maximum value of the solid phase volume fraction at the packing limit and was set to 0.63 as reference [57], collisional dissipation of solid fluctuating energy as shown in Eq. (13)

$${\gamma }_{s}=3\left(1-{e}^{2}\right){\varepsilon }_{s}^{2}{\rho }_{s}{g}_{0}{T}_{s}\left(\frac{4}{{d}_{p}}\sqrt{\frac{{T}_{s}}{\pi }}\right)$$
(13)

The transport equation for the granular temperature (or pseudothermal temperature) [58] as shown in Eq. (14)

$$\begin{array}{c}\frac{3}{2}\left[{\alpha }_{\text{s}}{\rho }_{\text{s}}\frac{\partial {T}_{\text{s}}}{\partial t}+{\alpha }_{\text{s}}{\rho }_{\text{s}}{U}_{\text{s},j}\frac{\partial {T}_{\text{s}}}{\partial {x}_{j}}\right]=2{\alpha }_{\text{s}}\left({\mu }_{\text{s},\text{ kin }}+{\mu }_{\text{s},\text{coll}}\right){S}_{\text{s},ij}{S}_{\text{s},ij} \\ +\frac{\partial }{\partial {x}_{j}}\left({k}_{\text{s}}\frac{\partial {T}_{\text{s}}}{\partial {x}_{j}}\right)-{\gamma }_{{T}_{\text{s}}}+{T}_{\text{gs}} \end{array}$$
(14)

where \({\mu }_{\text{s},\text{ kin}}\) is skin turbulent viscosity, \({\mu }_{\text{s},\text{coll}}\) is collision turbulent viscosity. And \({T}_{\text{gs}}\) is Transfer of kinetic energy of fluctuation form gas phase to solid. The expression for the collision frequency [58] as shown in Eq. (15)

$${\kappa }_{s}=\frac{150{\rho }_{s}{d}_{p}\sqrt{{T}_{s}\pi }}{384\left(1+e\right){g}_{0}}{\left[1+\frac{6}{5}{\varepsilon }_{s}{g}_{0}\left(1+e\right)\right]}^{2}+2{\rho }_{s}{\varepsilon }_{s}^{2}{d}_{p}(1+e){g}_{0}\sqrt{\frac{{T}_{s}}{\pi }}$$
(15)

Solid phase shear viscosity [58] as shown in Eq. (16)

$${\mu }_{s}={\mu }_{k\text{in}}+{\mu }_{col}$$
(16)

The skin viscosity as shown in Eq. (17)

$${\mu }_{kin}=\frac{4}{5}{\varepsilon }_{s}{\rho }_{s}{d}_{p}{g}_{0}\left(1+e\right)\sqrt{\frac{{T}_{s}}{\pi }}$$
(17)

The collision viscosity as shown in Eq. (18)

$${\mu }_{col}=\frac{10{\rho }_{s}{d}_{p}\sqrt{\pi {T}_{s}}}{96\left(1+e\right){g}_{0}{\varepsilon }_{s}}{\left[1+\frac{4}{5}{g}_{0}{\varepsilon }_{s}\left(1+e\right)\right]}^{2}$$
(18)

The drag force is given by [59] as shown in Eq. (19):

$$\overrightarrow{{F}_{D}}=\xi \left({\overrightarrow{U}}_{g}-{\overrightarrow{U}}_{s}\right)$$
(19)
$$\xi =\left\{\begin{array}{c}\frac{3}{4}{C}_{D}\frac{{\alpha }_{s}{\alpha }_{g}{\rho }_{g}\left|{\overrightarrow{\text{U}}}_{s}-{\overrightarrow{U}}_{g}\right|}{{d}_{s}}{\alpha }_{g}^{-2.65} \, {\text{F}}{\text{o}}{\text{r}} \, {\alpha }_{g}>0.8,\\ 150\frac{{\alpha }_{s}\left(1-{\alpha }_{g}\right){\mu }_{g}}{{\alpha }_{g}{d}_{s}^{2}}+1.75\frac{{\alpha }_{s}{\rho }_{g}\left|{\overrightarrow{U}}_{s}-{\overrightarrow{U}}_{g}\right|}{{d}_{s}} \, {\text{F}}{\text{o}}{\text{r}} \, {\alpha }_{g}\le 0.8,\end{array}\right.$$
$${\text{C}}_{\text{D}}=\frac{24}{{\alpha }_{g}R{e}_{s}}\left[1+0.15{\left({\alpha }_{g}R{e}_{s}\right)}^{0.687}\right]$$
$${\text{Re}}_{s}=\frac{{\rho }_{g}{d}_{s}\left|{\overrightarrow{U}}_{s}-{\overrightarrow{U}}_{g}\right|}{{\mu }_{g}}$$

The Saffman lift force due to presence of the particle in a shear flow is given based on the analytical results of Saffman [60] as shown in Eq. (20) and extended for higher Reynolds numbers according to Mei [61] as:

$${\overrightarrow{F}}_{L}=1.615{d}_{p}^{3} {\rho }_{g}{Re}_{s}^{-0.5}{C}_{SL}\left[\left({\overrightarrow{U}}_{s}-{\overrightarrow{U}}_{g}\right)\times {\overrightarrow{\omega }}_{r}\right]$$
(20)
$$ C_{SL} = \left\{ {\begin{array}{*{20}c} {\left( {1 - 0.3314\gamma^{0.5} } \right)e^{{\frac{{ - {\text{Re}}_{s} }}{10}}} + 0.3314\gamma^{0.5} } & {{\text{Re}}_{s} \le 40} \\ {0.0524( \gamma {\text{Re}}_{s} )^{0.5} } & {{\text{Re}}_{s} > 40} \\ \end{array} } \right. $$

\({\overrightarrow{\omega }}_{r}={\overrightarrow{\omega }}_{s}-{\overrightarrow{\omega }}_{g}\) is the relative angular velocity.

2.3 Boundary condition

The axial velocity of the gas phase was determined at the entrance of the pipe with a velocity profile obeying a fully developed 1/7th power law as shown in Eq. (21), while the radial velocity was set to zero. The inlet solid velocity same velocity of gas

$$\frac{{{\varvec{U}}}_{{\varvec{g}}}}{{{\varvec{U}}}_{\mathbf{C}}}={\left(1-\frac{{\varvec{r}}}{{\varvec{R}}}\right)}^\frac{1}{7}$$
(21)

where the \({U}_{\text{C}}\) is the center line velocity. As stated in [62], the turbulent kinetic energy as shown in Eq. (22) and dissipation rate as shown in Eq. (23):

$${k}_{g}=0.004{U}_{g,in}^{2}$$
(22)
$${{\varvec{\varepsilon}}}_{\mathbf{g}}=\frac{2{\left({{\varvec{k}}}_{\mathbf{g},\mathbf{i}\mathbf{n}}^{2}\right)}^\frac{3}{4}}{0.4187{\varvec{D}}}$$
(23)

According to [62, 63], the granular temperature (pseudothermal temperature) at the input was as follows Eq. (24):

$${{\varvec{T}}}_{{\varvec{S}}}=0.004{\left({{\varvec{U}}}_{\mathbf{s}}\right)}^{2}$$
(24)

The gas phase’s boundary condition at the wall specified no slip and no penetration while the particle phase's velocity normal to the wall was set to zero. However, the tangential velocity was determined by combining the momentum flux that was transmitted to the wall as a result of particle collisions with the wall and the particle phase shear stress that was present near to the wall. The tangential velocity at the wall shown in Eq. (25), the granular temperature at the wall in Eq. (26) as reference [64]. While the gas phase boundary condition at the wall taking as mention in [54]

$${{\varvec{U}}}_{\mathbf{s},\mathbf{w}}=-\frac{6{{\varvec{\mu}}}_{\mathbf{s}}{\boldsymbol{\alpha }}_{\mathbf{s},{\varvec{m}}{\varvec{a}}{\varvec{x}}}}{{\varvec{\pi}}{{\varvec{\rho}}}_{\mathbf{s}}{{\varvec{g}}}_{0}{\varvec{\phi}}{\boldsymbol{\alpha }}_{\mathbf{s}}\sqrt{3{{\varvec{T}}}_{\mathbf{s}}}}\frac{\partial {{\varvec{U}}}_{\mathbf{s},\mathbf{w}}}{\partial {\varvec{n}}}$$
(25)
$${{\varvec{k}}}_{\mathbf{s}}\frac{\partial {{\varvec{T}}}_{\mathbf{s}}}{\partial {\varvec{n}}}=\frac{\sqrt{3}{\varvec{\pi}}{{\varvec{\rho}}}_{\mathbf{s}}{\boldsymbol{\alpha }}_{\mathbf{s}}{{\varvec{g}}}_{0}\sqrt{{{\varvec{T}}}_{\mathbf{s}}}}{6{\boldsymbol{\alpha }}_{\mathbf{s},{\varvec{m}}{\varvec{a}}{\varvec{x}}}{{\varvec{\mu}}}_{\mathbf{s}}}\left[{\varvec{\phi}}{{\varvec{U}}}_{\mathbf{s},\mathbf{w}}^{2}-\frac{3{{\varvec{T}}}_{\mathbf{s}}}{2}\left(1-{{\varvec{e}}}_{\mathbf{p}\mathbf{w}}^{2}\right)\right]$$
(26)

where \({\varvec{\phi}}\) is the specularity coefficient representing the fraction of the particle phase momentum lost through collisions with the wall.

2.4 Solution procedure

By employing a hybrid technique to estimate the finite volume with a staggered grid and the SIMPLE algorithm to couple pressure and velocity, the partial differential equations will be solved. Convergence is considered to have occurred when the greatest residual sum for all components of the all solved variables (gas and solid) is less than 0.1%. Equation solving is done using house develop code by FORTRAN program. Line by line, the TDMA (tri-diagonal matrix algorithm) solution was utilized to calculate finite volume equations.

The diffusion terms employed the second-order accuracy of the central differencing scheme, whereas the convicted terms used a hybrid approach combining the first-order accuracy of the upwind differencing scheme with the second-order accuracy of the central differencing system. In general, these techniques provide sufficient accuracy, stability, and convergence. For pressure–velocity coupling, the SIMPLE method given by [54] was employed.

3 Results and discussions

3.1 Validation of code

The experiment [44] was used to examine the numerical code, and this experiment was used by many authors to validate their numerical code [22, 23, 53]. The validation process consisted of several key steps aimed at verifying the correctness and robustness of the implemented code. All validation had been checked for the grid-independent study, which used different grid sizes and different inlet velocity conditions (uniform and 1/7th power law). There is no different between the results with inlet different velocity profile due to the large of length for upstream pipe. So this effect not appear in this paper.

The comparison was made with the experiment by using different grid resolutions, which are\((400\times 100,700\times 130,802\times 150\;{\text {and}}\; 900\times 180)\). The instance being examined contains the geometry of a sudden expansion pipe with the upstream pipe radius (Rin = 0.025 m), step height (H = 0.035 m), and length of the downstream (L = 2.0 m), the inlet gas velocity 25.4 m/s. The density of particle 2500 kg/\({\text{m}}^{3}\) and the volume loading ratios is 0.005. The particle diameter is 50 µm. Figures display different stream-wise positions after the step wall of abrupt expansion. Figure (2) till (5) with corresponding X/H values of 1, 3.4, 7, and 14. X is the location measured from the step, and H is the step height. Figure 2 illustrates the axial velocity profile of the gas phase in a gas–solid turbulent flow. This figure provides valuable information about the grid-independent study, which it shows that the grid (\(700\boldsymbol{ }\times \boldsymbol{ }130\)) gives an acceptable result. As the grid resolution increase after that value, there is no notable change in velocity profile.

Fig. 2
figure 2

Axial velocity for gas phase in gas–solid turbulent flow with experiment [44] and with different grid resolution

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Figure 3 displays the axial velocity profile of the solid particles in a gas–solid turbulent flow. This figure provides important insights into the behavior and movement of the solid particles within the system. The axial velocity represents the velocity component along the direction of the flow for the solid particles, indicating how they are transported and distributed throughout the flow domain. The axial velocity profile depicted in the figure demonstrates how the velocity of the solid particles varies across the cross-sectional area of the flow. It shows the changes in velocity magnitude as the particles move through different regions, encountering gas flow, obstacles, and other particles. The profile may exhibit variations due to factors such as particle–particle interactions, gas drag, and turbulence.

Fig. 3
figure 3

Axial velocity for solid phase in gas–solid turbulent flow with experiment [44] and with different grid resolution

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There are two equations to predict the Reynolds stress (27), (28). Eq. (28) predicts a good accuracy for experiment data (11). Figure 4 presents a comparison between the predicted and measured normal Reynolds stresses. Additionally, the Reynolds stress in the mean direction \(\overline{{{\varvec{u}} }_{{\varvec{i}}}^{\mathbf{^{\prime}}}{{\varvec{u}}}_{{\varvec{j}}}^{\mathbf{^{\prime}}}}\) is calculated using a proposed anisotropic model [53]. The parameters \({{\varvec{c}}}_{{\varvec{\tau}}1}\) = 0.07 and \({{\varvec{c}}}_{{\varvec{\tau}}3}\)= 0.015, which are optimized and mentioned in [23, 54, 65], are unable to accurately anticipate the behavior of the Reynolds stress in the mean direction at Z/H = 1. This discrepancy can be attributed to the isotropic assumption inherent in the commonly used k-\({\varvec{\varepsilon}}\) model. However, the model shows good predictions for the location profiles at Z/H = 3.4, 7, and 14. In summary, the developed model demonstrates accurate forecasting capabilities for turbulent gas flow during sudden expansion, providing motivation for further theoretical research by the authors.

Fig. 4
figure 4

Comparison between computed square root of axial–axial fluid fluctuating velocity with experimental results of [44] at different stream-wise positions two-phase flow

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$$\overline{{u }_{i}{\prime}{u}_{j}{\prime}}=\frac{2}{3}k-\frac{{\mu }_{t}}{\rho }\left(\frac{2\partial {U}_{i}}{\partial {x}_{i}}\right)$$
(27)
$$\overline{{u }_{i}{\prime}{u}_{j}{\prime}} =\frac{2}{3}k+\frac{\left(2{c}_{\tau 1}-{c}_{\tau 3}\right)}{3}\frac{{k}^{3}}{{\varepsilon }^{2}}{\left(\frac{\partial u}{\partial y}\right)}^{2}$$
(28)

The prediction of Reynolds stress in the radial direction is fairly good, as shown in Fig. 5. In general, the developed model can predict the turbulent gas–solid flow through sudden expansion with accurate behavior, which encourages the authors to extend the theoretical study.

Fig. 5
figure 5

Comparison between computed square root of radial–radial fluid fluctuating velocity with experimental results of [44] at different stream-wise positions two-phase flow

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3.2 Effect of particle diameters

The parametric study examined with the same parameters as in the experiment [44] with different particle diameters. The area ratios is 5.76 and volume loading ratio is 0.01. A different particles diameter has a significant impact on gas–solid turbulent flow through a sudden expansion, affecting pressure, velocity, and local skin fraction. Larger particles, due to their higher inertia, contribute to decrease pressure. This elevated pressure is a result of the larger particles impeding the gas flow and causing greater disruption to the flow field. Additionally, the presence of large particles leads to a more pronounced decrease in gas velocity after the sudden expansion. As the gas velocity decreases, the large particles experience a greater deceleration and tend to disperse less. This localized accumulation of large particles can have implications for system performance, particle deposition, and overall flow behavior in gas–solid turbulent flows through sudden expansions.

Figure 6a illustrates the variation of the local skin friction coefficient with different particle diameters in gas–solid turbulent flow. When the particle diameters increase, the separation bubble decreases as shown in Fig. 9, this can be explained by the fact that as the particle diameter decreases, the availability of getting inside the separation bubble increases due to its light weight, which leads to following the fluid streamlines. As the particle enter the separation bubble helps to increase fraction coefficient. Figure 6b demonstrates that the pressure recovery is influenced by the particle diameter in gas–solid turbulent flow through a sudden expansion. As the particle diameter increases, the pressure decreases and separation bubble decreases [23].

Fig. 6
figure 6

The effect of particle diameters for local skin fraction and pressure recovery with AR = 5.76 and volume loading ratio = 0.01

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Figure 7 depicts the variation of gas velocity with respect to different particle diameters in gas–solid turbulent flow through a sudden expansion. The curves in the figure illustrate the relationship between velocity and particle diameter. When the particle diameter increases, the amount of turbulent kinetic energy decreases as shown in Fig. 10. Causing the energy transfer from particle to gas decreases, so the velocity will be decrease, also the particle velocity take a same trend as shown in Fig. 8. After the sudden directly the magnitude of the particle velocity is larger than gas velocity, this can be attributed by different in density, so the particles slowly effected by the area change rather than the gas. While as the particle goes far away from sudden section (X/H = 14) the particle loss the energy causing the particle velocity become less than gas velocity.

Fig. 7
figure 7

The axial velocity for gas phase for gas–solid turbulent flow with different particle diameter with AR = 5.76 and volume loading ratio = 0.01

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Fig. 8
figure 8

The axial velocity for solid phase for gas–solid turbulent flow with different particle diameter, with AR = 5.76 and volume loading ratio = 0.01

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3.2.1 Streamlines and turbulent kinetic energy

The small particles have less inertia than the large particle but the number of small particles is garter at the same volume loading ratios. Because of smaller inertia for the smallest particle, it is tend to follow the fluid path, so the particles get inside the separation bubble. The energy of these particles tends to form large separation bubbles, as shown in Fig. 9, which appeared in local skin fraction Fig. 6a.

Fig. 9
figure 9

Streamline for gas–solid turbulent flow with different particle diameter, the red lines tend to gas phase and blue lines tend to solid phase, with AR = 5.76 and volume loading ratio = 0.01

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The turbulent kinetic energy distribution within the gas phase in gas–solid turbulent flow through a sudden expansion is essential for understanding the flow behavior, energy exchange, and transport mechanisms in the gas phase. It provides insights into the intensity of turbulence and the distribution of energy within the gas phase, due to the larger surface area between the solid phase and gas phase causing a greater momentum change in presence of small particle as shown in Fig. 10

Fig. 10
figure 10

Particle diameter effects on the normalized turbulent kinetic energy \(\frac{k}{0.5*{uin}^{2}}\), with AR = 5.76 and volume loading ratio = 0.01

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3.3 Effect of solid void fraction

The void fraction plays a significant role in gas–solid turbulent flow through a sudden expansion. Void fraction refers to the ratio of the volume occupied by solid (voids) to the total volume of the flow domain. It quantifies the presence of solid particles within the gas phase and has important implications for the flow behavior and performance of the system.

The parametric study examined with same parameter as in experiment [44] with different void fraction and the area ratios is 5.76. The mean particle diameter is 50 µm s. In gas–solid turbulent flow through a sudden expansion, the void fraction affects several key aspects. Firstly, a higher void fraction indicates a higher concentration of solid particles within the flow. This increased particle concentration can lead to enhanced particle–particle and particle–wall interactions, resulting in intensified turbulence levels and altered flow patterns. The presence of solid particles disrupts the gas flow, causing deviations from the ideal behavior of gas flow in the absence of particles.

Figure 11a, Fig. 11b presents the variation of local skin friction and pressure recovery, respectively, for both solid particles and the gas phase with different void fractions in gas–solid turbulent flow through a sudden expansion. These figures provide valuable insights into the impact of void fraction on the skin friction and pressure recovery in the flow system.

Fig. 11
figure 11

The effect of void fraction for local skin fraction and pressure recovery, with AR = 5.76 and particle diameter 50 µm

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The relationship between void fraction and local skin friction as well as pressure recovery is complex and depends on various factors. Generally, an increase in the void fraction leads to higher local skin friction coefficients for both solid particles and the gas phase. This is because a higher void fraction corresponds to a higher concentration of solid particles, resulting in increased particle–fluid and particle–wall interactions, which in turn enhance the drag and skin friction. Regarding pressure recovery, the effect of void fraction can vary. In some cases, an increase in void fraction may lead to an increase in pressure recovery due to increased flow resistance caused by the presence of solid particles. However, the specific trend of pressure recovery with void fraction can also depend on other factors such as particle properties, flow conditions, and the geometry of the sudden expansion.

Figure 12 and Fig. 13 present the variation of axial velocity for the gas phase and axial velocity for solid particles, respectively, with different void fractions in gas–solid turbulent flow through a sudden expansion. These figures provide insights into the influence of void fraction on the axial velocities of both the gas phase and solid particles. As the void fraction increases, the amount of particles in domain increases causing a more energy exchange. The higher void fraction tend to effect as small particle diameter affected as shown in Fig. 7 and Fig. 8

Fig. 12
figure 12

The axial velocity for gas phase for gas–solid turbulent flow with different void fraction, with AR = 5.76 and particle diameter 50 µm

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Fig. 13
figure 13

The axial velocity for solid phase for gas–solid turbulent flow with different void fraction, with AR = 5.76 and particle diameter 50 µm

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Figure 14 presents the variation of void fraction with different values in gas–solid turbulent flow through a sudden expansion. The particles concentration just behind the sudden expansion section affected directly by the velocity, where the particles have a greater concentration parallel the smallest pipe radius. Then it decreases obviously near the wall. While, away from the sudden section the particles clearly distribution nearly constant as shown in (\(X/H= 9, 14\)).

Fig. 14
figure 14

Solid concentration with different inlet void fraction with AR = 5.76 and particle diameter 50 µm

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3.3.1 Streamlines and turbulent kinetic energy

Figure 15 presents the streamlines in gas–solid turbulent flow through a sudden expansion with different void fractions. Streamlines are imaginary lines that trace the paths followed by fluid particles within a flow field. By examining Figure, researchers can visualize the flow patterns and understand how different void fractions affect the movement and behavior of the gas and solid particles within the system. The figure provides insights into the impact of void fraction on the overall flow structure, including the formation of recirculation zones, flow separation, and the distribution of particles. When the void fraction increases, the separation will be increase. It is clearly appeared in local skin fraction (Fig. 11a).

Fig. 15
figure 15

Streamline for gas–solid turbulent flow with different void fraction, the red lines tend to gas phase and blue lines tend to solid phase with AR = 5.76 and mean particle diameter 50 µm

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Figure 16 shows the turbulent kinetic energy in gas–solid turbulent flow through a sudden expansion with different void fractions. Turbulent kinetic energy quantifies the energy associated with the turbulent motion of fluid particles. In gas–solid flows, turbulent kinetic energy is influenced by the presence of solid particles, which can enhance or dampen turbulence depending on their concentration and interactions with the gas phase. When the void fraction increases, the turbulent kinetic energy increases.

Fig. 16
figure 16

Void fraction effects on the normalized turbulent kinetic energy \(\frac{k}{0.5*{uin}^{2}}\) with AR = 5.76 and mean particle diameter 50 µm

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3.4 Effect of area ratios

After confirming the accuracy of the sudden expansion code, one significant factor that greatly influences fluid flow is the effect of area ratios. This investigation focuses on numerically studying the impact of area ratios (AR) on turbulent flow. Various area ratios were examined to observe their influence on streamlines, static pressure, and local skin friction coefficient. The conducted simulation use parameters as in the experiment described in references [44], but with a modification in the outer radius. The volume loading ratio is 0.01, and the particles diameters are 50 \(\mu m.\) Specifically, the outer radius (R) was set as the sum of the inner radius (Rin) and step height (H). To conduct the study, different area ratios were considered as shown in Table 1.

Table 1 Area ratios of sudden expansion
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The effect of area ratios in gas–solid turbulent flow in sudden expansion can significantly impact the flow behavior. When the area ratio is increased, there is a decrease in the gas velocity behind the step and an increase in the solid particle concentration. This leads to enhanced particle deposition and accumulation. Therefore, selecting an appropriate area ratio is crucial for optimizing gas–solid turbulent flow in sudden expansion scenarios. The effect of area ratios on pressure recovery in gas–solid turbulent flow in sudden expansion is significant. Increasing the area ratio helps to decrease the pressure recovery, and the separation bubble increase. Figure 17 presents the local skin fraction and pressure recovery, respectively. This can be explained as single-phase flow mentioned in [54, 66].

Fig. 17
figure 17

The effect area ratios for local skin fraction and pressure recovery, with mean particle diameter 50 µm and volume loading ratio 0.01

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3.4.1 Streamlines and turbulent kinetic energy

When the area ratios are increased in gas–solid turbulent flow through a sudden expansion, it can lead to an increase in flow separation, which in turn affects the streamlines and turbulent kinetic energy within the system. An increase in the area ratios typically results in a larger expansion of the flow passage. This expansion can create a more pronounced pressure difference between the upstream and downstream regions, leading to stronger flow separation. Flow separation occurs when the fluid detaches from the surface, forming recirculation zones and eddies. The presence of flow separation negatively impacts the streamlines. The separation zones disrupt the smooth flow patterns and cause deviations from the desired flow direction.

In terms of turbulent kinetic energy, an increase in area ratios can contribute to higher levels of turbulence due to the enhanced flow separation. The separated flow regions generate vortices and flow instabilities, increasing the intensity of turbulence and turbulent kinetic energy. This elevated turbulence can affect the particle transport and distribution within the flow and potentially leading to non-uniformity in gas–solid turbulent flows. However, in general, increasing the area ratios in gas–solid turbulent flow through a sudden expansion can result in heightened flow separation, distorted streamlines, and potentially increased turbulent kinetic energy levels, the streamline for two phase with different area ratios is shown in Fig. 18, and the normalized turbulent kinetic energy with different area ratios is shown in Fig. 19.

Fig. 18
figure 18

Streamline for gas–solid turbulent flow with different area ratios, the red lines tend to gas phrase and blue lines tend to solid phase, with particle diameter 50 µm mu and volume loading ratio 0.01

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Fig. 19
figure 19

Area ratios effects on the normalized turbulent kinetic energy \(\frac{k}{0.5*{uin}^{2}}\), with mean particle diameter 50 µm mu and volume loading ratio 0.01

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4 Conclusion

The numerical investigation of two-phase gas–solid turbulent flow using the Eulerian–Eulerian approach with variations in particle diameters, void fraction, and area ratios has provided valuable insights into the complex behavior of such systems. The study revealed that particle diameter significantly influences the flow dynamics. The void fraction affects the overall flow patterns and particle dispersion. Furthermore, these findings highlight the importance of considering particle diameter, void fraction, and area ratios in the design and optimization of gas–solid turbulent flows, enabling better control of system performance, particle deposition, and overall flow behavior. The numerical investigation serves as a foundation for further research and provides valuable insights for industrial applications involving gas–solid systems. From the current investigation, the following primary conclusions may be drawn:

  • Increasing particle diameters lead to pressure decreases, velocity decreases, and turbulent kinetic energy decreases.

  • The separation and eddies decrease as the particle diameters increase, so the streamlines reflect the impact of particle diameters in reattachment.

  • Increasing the solid void fraction leads to pressure increase, the velocity increase, and turbulent kinetic energy increase.

  • The separation and eddies increase as the void fraction increases.

  • Increasing area ratios helps to decrease pressure and increase turbulent kinetic energy

  • The separation and eddies increase as the area ratios increase.