Showing papers on "Flow separation published in 2014"

An improved dynamic non-equilibrium wall-model for large eddy simulation

Abstract: A non-equilibrium wall-model based on unsteady 3D Reynolds-averaged Navier-Stokes (RANS) equations has been implemented in an unstructured mesh environment. The method is similar to that of the wall-model for structured mesh described by Wang and Moin [Phys. Fluids 14, 2043–2051 (2002)], but is supplemented by a new dynamic eddy viscosity/conductivity model that corrects the effect of the resolved Reynolds stress (resolved turbulent heat flux) on the skin friction (wall heat flux). This correction is crucial in predicting the correct level of the skin friction. Unlike earlier models, this eddy viscosity/conductivity model does not have a stress-matching procedure or a tunable free parameter, and it shows consistent performance over a wide range of Reynolds numbers. The wall-model is validated against canonical (attached) transitional and fully turbulent flows at moderate to very high Reynolds numbers: a turbulent channel flow at Reτ = 2000, an H-type transitional boundary layer up to Reθ = 3300, and a hig...

The turbulent/non-turbulent interface and entrainment in a boundary layer

Abstract: The turbulent/non-turbulent interface in a zero-pressure-gradient turbulent boundary layer at high Reynolds number (Re = 14500) is examined using particle image velocimetry. An experimental set-up is utilized that employs multiple high-resolution cameras to capture a large field of view that extends 2δ× 1:1δ in the streamwise/wallnormal plane with an unprecedented dynamic range. The interface is detected using a criteria of local turbulent kinetic energy and proves to be an effective method for boundary layers. The presence of a turbulent/non-turbulent superlayer is corroborated by the presence of a jump for the conditionally averaged streamwise velocity across the interface. The steep change in velocity is accompanied by a discontinuity in vorticity and a sharp rise in the Reynolds shear stress. The conditional statistics at the interface are in quantitative agreement with the superlayer equations outlined by Reynolds (J. Fluid Mech., vol. 54, 1972, pp. 481-488). Further analysis introduces the mass flux as a physically relevant parameter that provides a direct quantitative insight into the entrainment. Consistency of this approach is first established via the equality of mean entrainment calculations obtained using three different methods, namely, conditional, instantaneous and mean equations of motion. By means of 'mass-flux spectra' it is shown that the boundary-layer entrainment is characterized by two distinctive length scales which appear to be associated with a two-stage entrainment process and have a substantial scale separation. © 2014 Cambridge University Press. ;

A dynamic slip boundary condition for wall-modeled large-eddy simulation

Abstract: Wall models for large-eddy simulation (LES) are a necessity to remove the prohibitive resolution requirements of near-wall turbulence in high Reynolds turbulent flows. Traditional wall models often rely on assumptions about the local state of the boundary layer (e.g., logarithmic velocity profiles) and require a priori prescription of tunable model coefficients. In the present study, a slip velocity boundary condition for the filtered velocity field is obtained from the derivation of the LES equations using a differential filter. A dynamic procedure for the local slip length is additionally formulated making the slip velocity wall model free of any a priori specified coefficients. The accuracy of the dynamic slip velocity wall model is tested in a series of turbulent channel flows at varying Reynolds numbers and in the LES of a National Advisory Committee for Aeronautics (NACA) 4412 airfoil at near-stall conditions. The wall-modeled simulations are able to accurately predict mean flow characteristics, including the formation of a trailing-edge separation bubble in NACA 4412 configuration. The validation cases demonstrate the effectiveness of this wall-modeling approach in both attached and separated flow regimes.

Confinement effects in shock wave/turbulent boundary layer interactions through wall-modelled large-eddy simulations

Abstract: We present wall-modelled large-eddy simulations (WLES) of oblique shock waves interacting with the turbulent boundary layers (TBLs) (nominal and ) developed inside a duct with an almost-square cross-section ( ) to investigate three-dimensional effects imposed by the lateral confinement of the flow. Three increasing strengths of the incident shock are considered, for a constant Mach number of the incoming air stream , by varying the height (1.1, 3 and 5 mm) of a compression wedge located at a constant streamwise location that spans the top wall of the duct at a 20° angle. Simulation results are first validated with particle image velocimetry (PIV) experimental data obtained at several vertical planes (one near the centre of the duct and three near one of the sidewalls) for the 1.1 and 3 mm-high wedge cases. The instantaneous and time-averaged structure of the flow for the stronger-interaction case (5 mm-high wedge), which shows mean flow reversal, is then investigated. Additional spanwise-periodic simulations are performed to elucidate the influence of the sidewalls, and it is found that the structure and location of the shock system, as well as the size of the separation bubble, are significantly modified by the lateral confinement. A Mach stem at the first reflected interaction is present in the simulation with sidewalls, whereas a regular shock intersection results for the spanwise-periodic case. Low-frequency unsteadiness is observed in all interactions, being stronger for the secondary shock reflections of the shock train developed inside the duct. The downstream evolution of secondary turbulent flows developed near the corners of the duct as they traverse the shock system is also studied.

Direct numerical simulation of pattern formation in subaqueous sediment

Abstract: We present results of direct numerical simulation of incompressible fluid flow over a thick bed of mobile spherically shaped particles. The algorithm is based upon the immersed-boundary technique for fluid–solid coupling and uses a soft-sphere model for the solid–solid contact. Two parameter points in the laminar flow regime are chosen, leading to the emergence of sediment patterns classified as ‘small dunes’, while one case under turbulent flow conditions leads to ‘vortex dunes’ with significant flow separation on the lee side. The wavelength, amplitude and propagation speed of the patterns extracted from the spanwise-averaged fluid–bed interface are found to be consistent with available experimental data. The particle transport rates are well represented by available empirical models for flow over a plane sediment bed in both the laminar and the turbulent regimes.

Observations of turbulent secondary flows in a rough-wall boundary layer

Abstract: The characteristics of a turbulent boundary layer overlying a complex roughness topography were explored with stereo particle-image velocimetry measurements in the wall-normal–spanwise plane. The roughness under consideration was replicated from a turbine blade damaged by deposition of foreign materials containing a broad range of topographical scales arranged in a highly irregular manner. The single-point turbulence statistics displayed strong spanwise heterogeneity, in particular spanwise-alternating low- and high-momentum flow pathways in the mean flow marked by enhanced Reynolds stresses and turbulent kinetic energy. The spanwise regions between high- and low-momentum flow pathways were occupied by swirling motions, suggesting the generation and sustainment of turbulent secondary flows due to the spanwise heterogeneity of the complex roughness under consideration. Similar observations were recently reported for more ordered spanwise roughness transitions by Nugroho, Hutchins & Monty (Intl J. Heat Fluid Flow vol. 41, 2013, pp. 90–102) and Willingham et al. (Phys. Fluids vol. 26, 2014, 025111).

Laminar, turbulent, and inertial shear-thickening regimes in channel flow of neutrally buoyant particle suspensions

Abstract: The aim of this Letter is to characterize the flow regimes of suspensions of finite-size rigid particles in a viscous fluid at finite inertia. We explore the system behavior as a function of the particle volume fraction and the Reynolds number (the ratio of flow and particle inertia to viscous forces). Unlike single-phase flows, where a clear distinction exists between the laminar and the turbulent states, three different regimes can be identified in the presence of a particulate phase, with smooth transitions between them. At low volume fractions, the flow becomes turbulent when increasing the Reynolds number, transitioning from the laminar regime dominated by viscous forces to the turbulent regime characterized by enhanced momentum transport by turbulent eddies. At larger volume fractions, we identify a new regime characterized by an even larger increase of the wall friction. The wall friction increases with the Reynolds number (inertial effects) while the turbulent transport is weakly affected, as in a state of intense inertial shear thickening. This state may prevent the transition to a fully turbulent regime at arbitrary high speed of the flow.

The formation mechanism and impact of streamwise vortices on NACA 0021 airfoil's performance with undulating leading edge modification

Abstract: Wings with tubercles have been shown to display advantageous loading behavior at high attack angles compared to their unmodified counterparts. In an earlier study by the authors, it was shown that an undulating leading-edge configuration, including but not limited to a tubercled model, induces a cyclic variation in circulation along the span that gives rise to the formation of counter-rotating streamwise vortices. While the aerodynamic benefits of full-span tubercled wings have been associated with the presence of such vortices, their formation mechanism and influence on wing performance are still in question. In the present work, experimental and numerical tests were conducted to further investigate the effect of tubercles on the flow structure over full-span modified wings based on the NACA 0021 profile, in the transitional flow regime. It is found that a skew-induced mechanism accounts for the formation of streamwise vortices whose development is accompanied by flow separation in delta-shaped regions near the trailing edge. The presence of vortices is detrimental to the performance of full-span wings pre-stall, however renders benefits post-stall as demonstrated by wind tunnel pressure measurement tests. Finally, primary and secondary vortices are identified post-stall that produce an enhanced momentum transfer effect that reduces flow separation, thus increasing the generated amount of lift.

Prediction of Transonic Buffet by Delayed Detached-Eddy Simulation

Abstract: A delayed detached-eddy simulation of the transonic buffet over a supercritical airfoil is performed. The turbulence modeling approach is based on a one-equation closure, and the results are compared to an unsteady Reynolds-averaged Navier–Stokes simulation using the same baseline model as well as experimental data. The delayed detached-eddy simulation successfully predicts the self-sustained unsteady shock-wave/boundary-layer interaction associated with buffet. When separation occurs, the flow exhibits alternate vortex shedding and a spanwise undulation. The method also captures secondary fluctuations in the boundary layer that are not predicted by unsteady Reynolds-averaged Navier–Stokes simulation. A map of flow separation emphasizes the differences between the delayed detached-eddy simulation and unsteady Reynolds-averaged Navier–Stokes flow topologies. Statistical pressure distributions and velocity profiles help assess the performance of each model. They indicate that the delayed detached-eddy simul...

Numerical simulation of sand waves in a turbulent open channel flow

Abstract: We develop a coupled hydro-morphodynamic numerical model for carrying out large-eddy simulation of stratified, turbulent flow over a mobile sand bed. The method is based on the curvilinear immersed boundary approach of Khosronejad et al. (Adv. Water Resour., vol. 34, 2011, pp. 829–843). We apply this method to simulate sand wave initiation, growth and evolution in a mobile bed laboratory open channel, which was studied experimentally by Venditti & Church (J. Geophys. Res., vol. 110, 2005, F01009). We show that all the major characteristics of the computed sand waves, from the early cross-hatch and chevron patterns to fully grown three-dimensional bedforms, are in good agreement with the experimental data both qualitatively and quantitatively. Our simulations capture the measured temporal evolution of sand wave amplitude, wavelength and celerity with good accuracy and also yield three-dimensional topologies that are strikingly similar to what was observed in the laboratory. We show that near-bed sweeps are responsible for initiating the instability of the initially flat sand bed. Stratification effects, which arise due to increased concentration of suspended sediment in the flow, also become important at later stages of the bed evolution and need to be taken into account for accurate simulations. As bedforms grow in amplitude and wavelength, they give rise to energetic coherent structures in the form of horseshoe vortices, which transport low-momentum near-bed fluid and suspended sediment away from the bed, giving rise to characteristic ‘boil’ events at the water surface. Flow separation off the bedform crestlines is shown to trap sediment in the lee side of the crestlines, which, coupled with sediment erosion from the accelerating flow over the stoss side, provides the mechanism for continuous bedform migration and crestline rearrangement. The statistical and spectral properties of the computed sand waves are calculated and shown to be similar to what has been observed in nature and previous numerical simulations. Furthermore, and in agreement with recent experimental findings (Singh et al., Water Resour. Res., vol. 46, 2010, pp. 1–10), the spectra of the resolved velocity fluctuations above the bed exhibit a distinct spectral gap whose width increases with distance from the bed. The spectral gap delineates the spectrum of turbulence from the low-frequency range associated with very slowly evolving, albeit energetic, coherent structures induced by the migrating sand waves. Overall the numerical simulations reproduce the laboratory observations with good accuracy and elucidate the physical phenomena governing the interaction between the turbulent flow and the developing mobile bed.

Large-scale contribution to mean wall shear stress in high-Reynolds-number flat-plate boundary layers up to 13650

Abstract: A numerical investigation of the mean wall shear stress properties on a spatially developing turbulent boundary layer over a smooth flat plate was carried out by means of a zonal detached eddy simulation (ZDES) technique for the Reynolds number range . Some asymptotic trends of global parameters are suggested. Consistently with previous findings, the calculation confirms the occurrence of very large-scale motions approximately to long which are meandering with a lateral amplitude of and which maintain a footprint in the near-wall region. It is shown that these large scales carry a significant amount of Reynolds shear stress and their influence on the skin friction, denoted , is revisited through the FIK identity by Fukagata, Iwamoto & Kasagi (Phys. Fluids, vol. 14, 2002, p. L73). It is argued that is the relevant parameter to characterize the high-Reynolds-number turbulent skin friction since the term describing the spatial heterogeneity of the boundary layer also characterizes the total shear stress variations across the boundary layer. The behaviour of the latter term seems to follow some remarkable self-similarity trends towards high Reynolds numbers. A spectral analysis of the weighted Reynolds stress with respect to the distance to the wall and to the wavelength is provided for the first time to our knowledge and allows us to analyse the influence of the largest scales on the skin friction. It is shown that structures with a streamwise wavelength contribute to more than of , and that those larger than still represent approximately of .

Experimental study of flow separation control on a low-Re airfoil using leading-edge protuberance method

Abstract: An experimental study of flow separation control on a low-Re c airfoil was presently investigated using a newly developed leading-edge protuberance method, motivated by the improvement in the hydrodynamics of the giant humpback whale through its pectoral flippers. Deploying this method, the control effectiveness of the airfoil aerodynamics was fully evaluated using a three-component force balance, leading to an effectively impaired stall phenomenon and great improvement in the performances within the wide post-stall angle range (22°–80°). To understand the flow physics behind, the vorticity field, velocity field and boundary layer flow field over the airfoil suction side were examined using a particle image velocimetry and an oil-flow surface visualization system. It was found that the leading-edge protuberance method, more like low-profile vortex generator, effectively modified the flow pattern of the airfoil boundary layer through the chordwise and spanwise evolutions of the interacting streamwise vortices generated by protuberances, where the separation of the turbulent boundary layer dominated within the stall region and the rather strong attachment of the laminar boundary layer still existed within the post-stall region. The characteristics to manipulate the flow separation mode of the original airfoil indicated the possibility to further optimize the control performance by reasonably designing the layout of the protuberances.

Skin-friction drag reduction in the turbulent regime using random-textured hydrophobic surfaces

Abstract: Technologies for reducing hydrodynamic skin-friction drag have a huge potential for energy-savings in applications ranging from propulsion of marine vessels to transporting liquids through pipes. The majority of previous experimental studies using hydrophobic surfaces have successfully shown skin-friction drag reduction in the laminar and transitional flow regimes (typically Reynolds numbers less than ≃106 for external flows). However, this hydrophobicity induced drag reduction is known to diminish with increasing Reynolds numbers in experiments involving wall bounded turbulent flows. Using random-textured hydrophobic surfaces (fabricated using large-length scalable thermal spray processes) on a flat plate geometry, we present water-tunnel test data with Reynolds numbers ranging from 106 to 9 × 106 that show sustained skin-friction drag reduction of 20%–30% in such turbulent flow regimes. Furthermore, we provide evidence that apart from the formation of a Cassie state and hydrophobicity, we also need a low surface roughness and an enhanced ability of the textured surface to retain trapped air, for sustained drag reduction in turbulent flow regimes. Specifically, for the hydrophobic test surfaces of the present and previous studies, we show that drag reduction seen at lower Reynolds numbers diminishes with increasing Reynolds number when the surface roughness of the underlying texture becomes comparable to the viscous sublayer thickness. Conversely, test data show that textures with surface roughness significantly smaller than the viscous sublayer thickness and textures with high porosity show sustained drag reduction in the turbulent flow regime. The present experiments represent a significant technological advancement and one of the very few demonstrations of skin-friction reduction in the turbulent regime using random-textured hydrophobic surfaces in an external flow configuration. The scalability of the fabrication method, the passive nature of this surface technology, and the obtained results in the turbulent regime make such hydrophobic surfaces a potentially attractive option for hydrodynamic skin-friction drag reduction.

Estimating the value of von Kármán’s constant in turbulent pipe flow

Abstract: Five separate data sets on the mean velocity distributions in the Princeton University/ONR Superpipe are used to establish the best estimate for the value of von Karman’s constant for the flow in a fully developed, hydraulically smooth pipe. The profiles were taken using Pitot tubes, conventional hot wires and nanoscale thermal anemometry probes. The value of the constant was found to vary significantly due to measurement uncertainties in the mean velocity, friction velocity and the wall distance, and the number of data points included in the analysis. The best estimate for the von Karman constant in turbulent pipe flow is found to be . A more precise estimate will require improved instrumentation.

Large-Eddy Simulations of a normal shock train in a constant-area isolator

Abstract: Large-eddy simulation of a normal shock train in a constant-area isolator model (M∞=161, Reθ≈1660) is carried out to investigate solution sensitivity with respect to a variety of physical modeling assumptions Simulations with spanwise periodic boundary conditions are first performed, the results of which are compared with experiment and validated with a three-level grid refinement study Due to the computational cost associated with resolving near-wall structures, the large-eddy simulation is run at a Reynolds number lower than that in the comparison experiment; thus, the confinement effect of the turbulent boundary layers is not exactly duplicated Although this discrepancy is found to affect the location of the first normal shock, the overall structure of the shock train and its interaction with the boundary layers matches the experiment quite closely Observations of pertinent physical phenomena in the experiment, such as a lack of reversed flow in the mean and the development of secondary shear laye

Unsteady forces on a circular cylinder at critical Reynolds numbers

Abstract: It is well known that the flow past a circular cylinder at critical Reynolds number combines flow separation, turbulence transition, reattachment of the flow, and further turbulent separation of the boundary layer. The transition to turbulence in the boundary layer causes the delaying of the separation point and an important reduction of the drag force on the cylinder surface known as the drag crisis. In the present work, large-eddy simulations of the flow past a cylinder at Reynolds numbers in the range 2.5 × 105-6.5 × 105 are performed. It is shown how the pressure distribution changes as the Reynolds number increases in an asymmetric manner, occurring first on one side of the cylinder and then on the other side to complete the drop in the drag up to 0.23 at Re = 6.5 × 105. These variations in the pressure profile are accompanied by the presence of a small recirculation bubble, observed as a small plateau in the pressure, and located around ϕ = 105∘ (measured from the stagnation point). This small recir...

Multiscale analysis of fluxes at the turbulent/non-turbulent interface in high Reynolds number boundary layers

Abstract: Analysis of fluxes across the turbulent/non-turbulent interface (TNTI) of turbulent boundary layers is performed using data from two-dimensional particle image velocimetry (PIV) obtained at high Reynolds numbers. The interface is identified with an iso-surface of kinetic energy, and the rate of change of total kinetic energy (K) inside a control volume with the TNTI as a bounding surface is investigated. Features of the growth of the turbulent region into the non-turbulent region by molecular diffusion of K, viscous nibbling, are examined in detail, focussing on correlations between interface orientation, viscous stress tensor elements, and local fluid velocity. At the level of the ensemble (Reynolds) averaged Navier-Stokes equations (RANS), the total kinetic energy K is shown to evolve predominantly due to the turbulent advective fluxes occurring through an average surface which differs considerably from the local, corrugated, sharp interface. The analysis is generalized to a hierarchy of length-scales by spatial filtering of the data as used commonly in Large-Eddy-Simulation (LES) analysis. For the same overall entrainment rate of total kinetic energy, the theoretical analysis shows that the sum of resolved viscous and subgrid-scale advective flux must be independent of scale. Within the experimental limitations of the PIV data, the results agree with these trends, namely that as the filter scale increases, the viscous resolved fluxes decrease while the subgrid-scale advective fluxes increase and tend towards the RANS values at large filter sizes. However, a definitive conclusion can only be made with fully resolved three-dimensional data, over and beyond the large dynamic spatial range presented here. The qualitative trends from the measurement results provide evidence that large-scale transport due to the energy-containing eddies determines the overall rate of entrainment, while viscous effects at the smallest scales provide the physical mechanism ultimately responsible for entrainment. Data spanning over a decade in Reynolds number suggest that the fluxes (or the entrainment velocity) scale with the friction velocity (or equivalently the local turbulent fluctuating velocity), whereas Taylor microscale and boundary-layer thickness are the appropriate length scales at small and large filter sizes, respectively.

Fluid flow analysis of a shark-inspired microstructure

Abstract: The scales of fast-swimming sharks contain riblet structures with microgrooves, aligned in the direction of fluid flow, that result in water moving efficiently over the surface. In previous studies, these riblet structures have shown a drag reduction of up to 10 % when compared with a smooth, flat surface. These studies have suggested two prevalent drag-reduction mechanisms which involve the effect of vortices and turbulence fluctuations. To further explore relevant mechanisms and study the effect of riblet geometry and flow properties on drag, vortices and turbulence fluctuations, various shark-skin-inspired riblet structures were created using computational models in which velocity, viscosity, spacing, height and thickness parameters were independently modified. A relevant mechanism of drag reduction is discussed to relate riblet parameters and flow properties to drag change and vortex size. Modelling information will lead to a better understanding of riblets and allow for optimum drag-reducing designs for applications in marine, medical and industrial fields.

Flow control of an oblique shock wave reflection with micro-ramp vortex generators: Effects of location and size

Abstract: This study investigates the influences of micro-ramp size and location on its effectiveness as a flow control device for oblique shock wave reflections The effectiveness is measured in terms of the size of the shock-induced separation bubble and the reflected shock unsteadiness Particle image velocimetry measurements were carried out on the interaction region and the mixing region between micro-ramp and interaction The separation bubble is shown to be most sensitive to the momentum flux contained in the lower 43% of the incoming boundary layer The momentum flux added to this region scales linearly with micro-ramp height and larger micro-ramps are shown to be more effective in stabilizing the interaction Full boundary layer mixing is attained 57δ downstream of the micro-ramp and this forms a lower limit on the required distance between micro-ramp and the start of the interaction region Typical reductions in the average separated area and the shock unsteadiness of 87% and 51%, respectively, were recorded Results, however, depend strongly upon the spanwise location, with the micro-ramp being most effective along its centerline

The quiescent core of turbulent channel flow

Abstract: The identification of uniform momentum zones in wall-turbulence, introduced by Adrian, Meinhart & Tomkins (J. Fluid Mech., vol. 422, 2000, pp. 1–54) has been applied to turbulent channel flow, revealing a large ‘core’ region having high and uniform velocity magnitude. Examination of the core reveals that it is a region of relatively weak turbulence levels. For channel flow in the range , it was found that the ‘core’ is identifiable by regions bounded by the continuous isocontour lines of the streamwise velocity at (95 % of the centreline velocity). A detailed investigation into the properties of the core has revealed it has a large-scale oscillation which is predominantly anti-symmetric with respect to the channel centreline as it moves through the channel, and there is a distinct jump in turbulence statistics as the core boundary is crossed. It is concluded that the edge of the core demarcates a shear layer of relatively intense vorticity such that the interior of the core contains weakly varying, very low-level turbulence (relative to the flow closer to the wall). Although channel flows are generally referred to as ‘fully turbulent’, these findings suggest there exists a relatively large and ‘quiescent’ core region with a boundary qualitatively similar to the turbulent/non-turbulent interface of boundary layers, jets and wakes.

Computational and Experimental Analysis of a High-Performance Airfoil Under Low-Reynolds-Number Flow Condition

Abstract: A high-performance Ishii airfoil was analyzed using both a wind-tunnel and large-eddy simulations at a low-Reynolds-number condition (Re=23,000). The design guidelines for an airfoil shape with a high lift-to-drag ratio under the aforementioned condition are described by analyses of flowfields and aerodynamic characteristics of the Ishii airfoil. Compared with conventional airfoils, such as the NACA 0012 and NACA 0002, the shape characteristic effects of the Ishii airfoil on its flowfield and aerodynamic characteristics are discussed. The shape on the suction side of the Ishii airfoil can cause delays in the flow separation at low angle of attacks. The separated flow reattaches, and a separation bubble forms even when trailing-edge separation changes to leading-edge separation. The separation bubble contributes to an increase in lift coefficient. In addition, the Ishii airfoil can gain a high positive pressure on the pressure side as compared with the other two symmetric airfoils due to the camber near th...

Numerical Simulation of Forced Convective Heat Transfer Past a Square Diamond-Shaped Porous Cylinder

Abstract: Fluid flow and heat transfer around and through a porous cylinder is an important issue in engineering applications. In this paper a numerical study is carried out for simulating the fluid flow and forced convection heat transfer around and through a square diamond-shaped porous cylinder. The flow is two-dimensional, steady, and laminar. Conservation laws of mass, momentum, and heat transport equations are applied in the clear region and Darcy–Brinkman–Forchheimer model for simulating the flow in the porous medium has been used. Equations with the relevant boundary conditions are numerically solved using a finite volume approach. In this study, Reynolds and Darcy numbers are varied within the ranges of $$1