Critical Reynolds Number Research Articles

Experimental investigation on transitional flow of supercritical hydrocarbon fuel within straight channels of Printed Circuit Heat Exchangers for hypersonic thermal management systems

The onboard supercritical CO2 closed Brayton cycle (SCBC) has established an effective thermal management pathway for hypersonic vehicles operating at high Mach numbers. A Printed Circuit Heat Exchanger (PCHE) is employed within the cycle system to facilitate heat transfer between fuel and carbon dioxide, with its thermo-hydraulic characteristics significantly impacting both power generation and propulsion performance. Given the insufficient research on the flow and heat transfer of hydrocarbon fuels in PCHEs, this study aims to experimentally address the gaps in understanding transition flow for supercritical hydrocarbon fuels, particularly within semi-circular microchannels. We conducted experiments on the flow transitions using a specially designed test piece with a diameter of 1 mm, replicating the PCHE channel characteristics. The experimental Reynolds number range from 300 to 8300 spans laminar, transition, and turbulent flow regimes. Friction factors for n-decane are obtained under adiabatic and heated conditions. The results validate the nomenclature of the quasi-turbulent region and reveal that the critical Reynolds numbers for the transitional, quasi-turbulent and hydraulic smooth turbulent regions under adiabatic conditions are 2200, 2440 and 2980, respectively. A reduction in friction drag attributed to boundary layer viscosity is observed to alter the friction factor under heating conditions, while heating influences both the onset and termination of the transition region, but does not affect its width. Classical correlations are compared with the experimental results, and new empirical correlations for both adiabatic and non-adiabatic conditions are developed, showing superior predictive accuracy compared to existing models. This provides a reliable tool for the engineering design of heat exchangers in the onboard thermal manage systems.

Unraveling the corrosion behavior and corrosion scale evolution of N80 steel in high-temperature CO2 environment: The role of flow regimes

During CO2 transportation and storage, metal equipment such as oilfield pipelines suffers from severe CO2 corrosion, especially in harsh downhole injection equipment. In this study, we investigated the corrosion behavior of oil well tubing in a high-temperature, high-pressure (HTHP) CO2-containing environment. The evolution of the corrosion scale was also examined under different flow regimes. The results reveal a lower corrosion rate at 150 °C compared to 80 °C under different flow regimes, with localized corrosion intensifying as temperature and rotational speeds (vrs) increase. The temperature also induces the corrosion scale conversion of aragonite-type CaCO3 (80 °C) to calcite-type CaCO3 (150 °C). Specifically, the variation of the corrosion rate and the corrosion scale evolution can be attributed to the vortices within the reactor. The intact vortex cells enhance mass transfer while also promoting nucleation and growth of CaCO3. However, when vrs exceeds the critical Reynolds number, the vortex cells are disrupted, resulting in viscous dissipation and a reduced corrosion rate.

Examining the Role of Surface Temperature on Boundary Layer Stability and Transition to Turbulence

This study investigates the influence of varying surface temperature on the critical Reynolds number to understand the onset of turbulence in the boundary layer of a flat plate. Computational Fluid Dynamics (CFD) simulations in ANSYS Fluent have been utilized for the purpose of this study. The study systematically varies surface temperature to measure its impact on flow stability. The research demonstrates that an increase in surface temperature results in a lower critical Reynolds number (leading to an earlier transition from laminar to turbulent flow) by modeling air as the working fluid and applying the k-ω SST turbulence model. The results align with theoretical predictions which highlight the destabilizing effect of temperature-induced viscosity changes. This finding has practical implications for aerospace engineering, HVAC systems, and environmental modeling where controlling turbulence can optimize performance and energy efficiency. The study acknowledges limitations due to the use of a 2D model and suggests future work using 3D simulations and other flow conditions to expand the applicability of the findings.

Turbulence model adaptability at critical Reynolds numbers and applications in wake control via fairings

The water-drop shaped fairings with varying shape angles are attached to a circular cylinder to achieve wake control and vortex suppression at critical Reynolds numbers. To ensure the capability of Reynolds averaged Navier–Stokes (RANS), detached eddy simulation (DES) and large eddy simulation (LES) models at the critical Reynolds number region, three representative turbulence models are employed: LES with σ subgrid-scale (SGS) model, delayed DES model with improved wall-modeling capability (IDDES) and shear stress transport (SST) k−ω RANS model. These models are utilized to simulate flow around a circular cylinder at Reynolds number Re=2.5×105. The solver used in this paper is further developed based on the high-resolution algorithm platform for incompressible flow (HRAPIF). The comparative analysis of the results from the three turbulence models has been rigorously validated and investigated. An exhaustive examination of the mean flow field, Reynolds stresses, characteristic lengths, and instantaneous flow fields among the models reveals instructive insights. The IDDES and σ-LES models predict the hydrodynamic forces, the so-called ‘drag crisis’, alongside the pressure distribution and skin friction coefficient with high precision. The σ-LES model stands out for its superior accuracy, while the IDDES model is also a viable alternative, offering commendable accuracy with a reduced demand for mesh density. Subsequently, the IDDES model is selected for wake control calculations using fairings with five distinct shape angles (30∘,45∘,60∘,75∘ and 90∘) at Re=2.5×105. In-depth comparisons to the bare cylinder and subcritical results reveal that the wake control effect varies at the critical Reynolds region. The fairing with a 30° shape angle substantially suppresses hydrodynamic forces. The lift coefficient experiences a remarkable decrease of approximately 96%, while the drag coefficient diminishes by about 90%. Concurrently, fairings with angles from 45° to 90° lead to reductions in drag coefficient of 11.6%, 10%, 3% and 4%, respectively. A 75% lowering in the lift force coefficient is found even with a short fairing with shape angle α=90∘, which may be a meaningful instruction to industrial design.

Bouncing behaviour of a particle settling through a density transition layer

The present work focuses on a specific bouncing behaviour as a spherical particle settles through a density interface in the absence of a neutral buoyant position. This behaviour was initially discovered by Abaid et al. (Phys. Fluids, vol. 16, issue 5, 2004, pp. 1567–1580) in salinity-induced stratification. Both experimental and numerical investigations are conducted to understand this phenomenon. In our experiments, we employ particle image velocimetry (PIV) to measure the velocity distribution around the particle and to capture the transient wake structure. Our findings reveal that the bouncing process begins after the wake detaches from the particle. The PIV results indicate that an upward jet forms at the central axis behind the particle following wake detachment. By performing a force decomposition procedure, we quantify the contributions from the buoyancy of the wake ( $F_{sb}$ ) and the flow structure ( $F_{sj}$ ) to the enhanced drag. It is observed that $F_{sb}$ contributes primarily to the enhanced drag at the early stage, whereas $F_{sj}$ plays a critical role in reversing the particle's motion. Furthermore, our results indicate that the jet is a necessary condition for the occurrence of the bouncing motion. We also explore the minimum velocities (where negative values denote the occurrence of bouncing) of the particle, while varying the lower Reynolds number $Re_l$ , the Froude number $Fr$ , and the upper Reynolds number $Re_u$ , within the ranges $1 \leqslant Re_l\leqslant 125$ , $115 \leqslant Re_u\leqslant 356$ and $2 \leqslant Fr\leqslant 7$ . Our findings suggest that the bouncing behaviour is influenced primarily by $Re_l$ . Specifically, we observe that the bouncing motion occurs below a critical lower Reynolds number $Re^\ast _{l}=30$ in our experiments. In the numerical simulations, the highest value for this critical number is $Re^\ast _{l}=46.2$ , which is limited to the parametric ranges studied in this work.

Aerodynamic and acoustic evaluation of a cylinder covered by stretched grooved fabric near critical Reynolds numbers

This study investigates aerodynamic drag and acoustic noise generated from a circular cylinder by wrapping longitudinally grooved fabrics. The fabrics are stretched to different levels to explore their potential for flow and acoustic control near critical Reynolds numbers. To analyze the fabrics' surface characteristics, 3D scanning and several microscopic techniques were employed. The parallel-mounted load cells and an array of microphones were used to measure the drag and noise at Reynolds numbers, based on cylinder diameter, ranging from \num{3.3e4} to \num{1.1e5} in an anechoic wind tunnel. Additionally, hotwire anemometry was used to examine the wake structures in the vicinity of the cylinder. The surface scanning results revealed that stretching the fabric transversely to twice its original size reduced the groove depth by approximately $54\%$ and increased the width by around $25\%$, resulting in an overall reduction of the arithmetic surface roughness $S_a$ by $18.6\%$. Based on the wind tunnel data, it was observed that a grooved fabric with higher stretch, and thus lower surface roughness exhibited a minimum drag coefficient at a higher Reynolds number, along with reduced acoustic tonal noise levels.

Research on gas-liquid interface parameters related to thermal performance of frost-free evaporator of air source heat pump

Direct spray frost-free air source heat pumps (FFASHPs) base on liquid desiccant solutions are a more promising renewable energy technology for developing net-zero emission buildings. In order to improve the thermal performance of the frost-free evaporator, a numerical mode was established based on the penetration theory and the two-film theory to reveal the interfacial heat and mass transfer mechanism of LiCl solution falling film absorption on the air side of the frost-free evaporator. The distribution characteristics of temperature, water vapor concentration, effective diffusion coefficient, morphology, velocity and pressure at the interface under different air Reynolds numbers and temperatures were analyzed. The results show that the distribution uniformity of interface parameters has a greater effect on heat transfer performance than its average value. The critical Reynolds number is 391.0 under the present study, and the distribution uniformity of interface parameters and the intensity of interface fluctuation are improved, and the thermal performance reaches the peak. The COP of FFASHP system can be improved effectively when the frost-free evaporator is operated under the critical condition matching its parameters. The purpose of the study is to provide theoretical support for the performance improvement and efficient operation of frost-free evaporators.

A numerical study on the issue of Reynolds independence of flow and dispersion within isolated street canyons

Achieving Reynolds number independence is a prerequisite for conducting scale model experiments. This paper presents a numerical model of the two-dimensional isolated street canyon, utilizing the validated reliable k-epsilon turbulence model based on wind tunnel experimental data, to investigate the influence of approaching wind profiles (α) and windward building width (WB) on achieving Reynolds number independence in the isolated street canyon and compare these characteristics with those observed in urban street canyon. quantitative criteria, including revised relative change ratios (RRCs) ≤ 20% and dimensionless concentration relative difference (K_RD) ≤ 5%, were employed in conjunction with velocity, concentration contours, and streamlines to determine the corresponding critical Reynolds numbers. The results suggest that variations in approaching wind profiles primarily influence the flow field structure within the isolated street canyon after achieving Reynolds number independence, particularly impacting the scale of the counterclockwise vortex. However, these variations do not significantly alter the critical Reynolds number (Recrit) required for achieving Re-independence, which is estimated to be Recrit = 2.2×10⁴ for the isolated street canyon. Varying WB affects the formation of counterclockwise vortex within the isolated street canyon. A reduction in WB facilitates its formation and accelerates reaching Re-independence. The Recrit is found to be linearly proportional to WB variation. The flow field structure in urban street canyons remains relatively stable, with a recommended Recrit = 3.1×10⁴. When WB is 18 cm, the Recrit for the isolated street canyon aligns with that of the urban street canyon.

Flow of fluids with pressure-dependent viscosity down an incline: Long-wave linear stability analysis

In this paper, we investigate the linear stability of a gravity-driven fluid with pressure-dependent viscosity flowing down an inclined plane. The linear stability analysis is formulated using the long-wave approximation method. We show that the onset of instability occurs at a critical Reynolds number that depends on the material and geometrical parameters. Our results suggest that the dependence of the viscosity on pressure can influence the stability characteristics of the flow down an incline.

Drawing of fibres composed of shear-thinning or shear-thickening fluid with internal holes

We explore the drawing of a shear-thinning or shear-thickening thread with an axisymmetric hole that evolves due to axial drawing, inertia and surface tension effects. The stress is assumed to be proportional to the shear rate raised to the $n$ th power. The presence of non-Newtonian rheology and surface tension forces acting on the hole introduces radial pressure gradients that make the derivation of long-wavelength equations significantly more challenging than either a Newtonian thread with a hole or shear-thinning and shear-thickening threads without a hole. In the case of weak surface tension, we determine the steady-state profiles. Our results show that for negligible inertia the hole size at the exit becomes smaller as $n$ is decreased (i.e. strong shear-thinning effects) above a critical draw ratio, but surprisingly the opposite is true below this critical draw ratio. We determine an accurate estimate of the critical draw ratio and also discuss how inertia affects this process. We further show that the dynamics of hole closure is dominated by a different limit, and we determine the asymptotic forms of the hole closure process for shear-thinning and shear-thickening fluids with inertia. A linear instability analysis is conducted to predict the onset of draw resonance. We show that increased shear thinning, surface tension and inlet hole size all act to destabilise the flow. We also show that increasing shear-thinning effects reduce the critical Reynolds number required for unconditional stability. Our study provides valuable insights into the drawing process and its dependence on the physical effects.

Numerical Approach of the First Instability Appearance in Inclined Taylor–Couette System

The present study numerically investigates the three-dimensional forced and mixed convection heat transfer in an inclined Taylor–Couette system (η=0.5 and Γ=20) submitted to a radial temperature gradient. The main objective of this study is to determine the effect of the angle inclination duct on the thermal and dynamic fields. Several cases have been dealt with depending on the inclination angle to detect the critical Reynolds number in each case. The model of the conservation equations with their boundary conditions is numerically solved by the finite volume method with a second-order spatiotemporal discretization. The results show that, in forced convection, the effect of the inclination angle is inexistent on the velocity field. However, with the presence of buoyancy effects, which impact flow stability and the transition to turbulence, the inclination influences both velocity and temperature fields. It also shows that selecting the vertical position of the annulus is preferable to obtain hydrodynamic stability in mixed convection. At the same time, from the thermal point of view, it is preferable to select the horizontal position to get dynamic and thermal stability.

Can boundary slip destabilize rotating microchannel flows?

Deviation from the traditional no-slip boundary condition due to factors like surface roughness and wettability is of paramount importance in microfluidics and nanofluidics, as it is attributable to its significance in drag reduction, flow control and enhancement and improved mixing. Augmentation in mixing, in turn, is known to strongly correlate with potential instabilities in the flow structure. Reported research studies indicate that slip is an inherent flow stabilizer in microfluidics, to the extent that with sufficient slip, the flow becomes linearly stable against all wavelike disturbances for all wavelengths and Reynolds numbers [“The linear stability of slip channel flows,” Phys. Fluids 34,074103(2022)]. Contrary to such intuitive proposition, here we show that slip effects can destabilize microchannel flows under spanwise rotation, delving on the interplay of rotational forces and slippery hydrodynamics. Our results reveal that increasing the slip length decreases the critical rotation speed, indicating lower rotational effort required to destabilize the flow, whereas the critical Reynolds number for the flow remains effectively unaltered for different slip lengths in a spanwise rotating system. As the slip length increases progressively, the critical rotation number (dimensionless rotational speed) for the onset of instability decreases further, then remains constant up to a certain limit, and subsequently declines with additional enhancement in the slip length. This indicates the potential for deploying customized hydrophobic (slippery) substrates to facilitate transitions from stable to unstable modes by simple tuning of the rotational speed—a paradigm that offers great promise in various applications ranging from materials synthesis to biomedical technology.

Mathematical modeling of sound propagation through the respiratory tract in a modified gas environment

The aim of the work was to develop a mathematical model of sound propagation through the human respiratory tract when breathing certain respiratory gas mixtures. We based the model on an assembler model of branching of the respiratory tract. For calculations, we presented the human tracheobronchial tree in the form of an equivalent T-shaped cut section diagram, which includes parameters of the walls of the respiratory tract. We carried the simulation out in two main stages. At the first stage, we calculated the impedance of the flow from the small airways to the larger airways. At the second stage, we calculated the sound pressure caused by breathing. As a result, we obtained the curves of the impedance dependence on the frequency spectrum, as well as the sound power from the respiratory flow. We noted that when breathing air, the impedance value remains constant with increasing frequency. Moreover, we also noted that when exposed to a heavier gas than air, the impedance value constantly increases, and when exposed to lighter gases, after reaching a maximum, the impedance value decreases. We found that the highest sound pressure is typical for the oxygen-krypton mixture, and the lowest for the oxygen–helium mixture. We also found that in the case of modeling the laminar flow regime (with a critical Reynolds number equal to 1800), sound affects how many generations of bronchi more than in the case of modeling the transient flow regime (with a critical Reynolds number equal to 2700). Thus, we drew conclusions about the influence of the physical properties of gaseous media on the amount of sound pressure and the propagation of sound through the human respiratory tract.

Mapping vortex-induced forces of oscillating bluff bodies from subcritical to critical Reynolds numbers

Mapping vortex-induced forces of oscillating bluff bodies from subcritical to critical Reynolds numbers

Subcritical transitional flow in two-dimensional plane Poiseuille flow

Recently, subcritical transition to turbulence in the quasi-two-dimensional (quasi-2-D) shear flow with strong linear friction (Camobreco et al., J. Fluid Mech., vol. 963, 2023, R2) has been demonstrated by the 2-D mechanism at $Re = 71\,211$ , and the nonlinear Tollmien–Schlichting (TS) waves related to the edge state were approached independently of initial optimal disturbances. For 2-D plane Poiseuille flow, transition to the fully developed turbulence requires that the Reynolds number is several times larger than the critical Reynolds number $Re_c$ (Markeviciute & Kerswell, J. Fluid Mech., vol. 917, 2021, A57). In this paper, we observed the subcritical transitional flow in 2-D plane Poiseuille flow driven by the nonlinear TS waves by both linear and nonlinear optimal disturbances ( $Re < Re_c$ ) with different quantitative edge states. The nonlinear optimal disturbances could trigger the sustained subcritical transitional flow for $Re \geqslant 2400$ . The initial energy for nonlinear optimal disturbance is more efficient than the linear optimal disturbance in reaching the subcritical transitional flow for $2400 \leqslant Re \leqslant 5000$ . Moreover, the initial energy of linear optimal disturbance is larger than the energy of its edge state. The nonlinear TS waves along the edge state are formed by the nonlinear optimal disturbances to trigger transitional flow, which agrees well with the main conclusions of Camobreco et al. (J. Fluid Mech., vol. 963, 2023, R2), while the required $Re$ of 2-D plane Poiseuille flow is much smaller.

Effects of Reynolds number on the unsteady performance and flow of a pre-swirl stator pump-jet propulsor

ABSTRACT Reynolds number is important when experimenting or numerical predicting an underwater propulsor. In order to assess the Reynolds effects on numerical prediction, this paper considers the performance and flow of a pre-swirl stator pump-jet propulsor (PJP) under different working points, where the Reynolds number is controlled by varying the oncoming flow velocity and rotating speed. Numerical predictions are conducted through a hybrid turbulence modelling approach, named the embedded large eddy simulation, whose predictions of performance and flow of the PJP show good agreements with experiments. Then, the performance, force components, flow, and vortical systems around the PJP are discussed and analysed. Results indicate that not only the propulsion performance but also the unsteady fluctuating forces and flow are considerably affected by the Reynolds number. The performance is slightly affected after a Reynolds number higher than two times the critical Reynolds number suggested by the International Towing Tank Conference (ITTC). More importantly, the stator wake is highly affected and causes high fluctuations of thrust and torque of the rotor when the Reynolds number is below two times the critical Reynolds number. This work provides some new insights into the numerical calculating of PJP hydrodynamics and contributes to exploring the unsteady characteristics and rotor-stator interaction.

Sediment size effects on non-Darcy flow: insights from Izbash equation and Forchheimer inertial coefficient analysis

AbstractThe transition from Darcy to non-Darcy flow regimes was investigated using column experiments. This revealed key relationships between sediment characteristics and critical thresholds for the onset of the non-Darcy flow regime, as well as inertial flow parameters. An exponential dependence of the critical Reynolds number (Rec) on hydraulic conductivity (K) and a linear dependence on sediment size (d50) was found. The analysis revealed a potentially universal relationship between the critical hydraulic gradient (Ic) and K, with a power-law exponent of –3/2, consistent with previous investigations. Additionally, Ic was found to be inversely proportional to the power law of the square root of d50. Novel relationships are derived for estimating the Izbash equation inertial exponent (n) and the Forchheimer inertial coefficient ($$\beta$$ β ) based on sediment characteristics. The exponent (n) was found to decrease with d50 and increase with K, following power-law relationships. A new equation is proposed, capable of predicting $$\beta$$ β with slightly improved accuracy, outperforming numerous and previously proposed empirical equations. Additionally, these data validate the works of Ergun and Irmay as an alternative for $$\beta$$ β estimation using porosity and sediment size. As the attainment of statistical significance in multiparameter curve fitting can be trivial, it has led to the proliferation of empirical equations for estimating β. This study highlights the limitations of existing empirical methods in determining β and emphasizes the necessity for a universal approach to predict this critical parameter, which will facilitate broader adoption of the Forchheimer equation.

Vorticity and magnetic dynamo from subsonic expansion waves. II. Dependence on the magnetic Prandtl number, forcing scale, and cooling time

The amplification of astrophysical magnetic fields takes place via dynamo instability in turbulent environments. Vorticity is usually present in any dynamo, but its role is not yet fully understood. This work is an extension of previous research on the effect of an irrotational subsonic forcing on a magnetized medium in the presence of rotation or a differential velocity profile. We aim to explore a wider parameter space in terms of Reynolds numbers, the magnetic Prandtl number, the forcing scale, and the cooling timescale in a Newtonian cooling. We studied the effect of imposing that either the acceleration or the velocity forcing function be curl-free and evaluated the terms responsible for the evolution vorticity. We used direct numerical simulations to solve the fully compressible, resistive magnetohydrodynamic equations with the Pencil Code. We studied both isothermal and non-isothermal regimes and addressed the relative importance of different vorticity source terms. We report no small-scale dynamo for the models that do not include shear. We find a hydro instability, followed by a magnetic one, when a shearing velocity profile is applied. The vorticity production is found to be numerical in the purely irrotational acceleration case. Non-isothermality, rotation, shear, and density-dependent forcing, when included, contribute to increasing the vorticity. As in our previous study, we find that turbulence driven by subsonic expansion waves can amplify the vorticity and magnetic field only in the presence of a background shearing profile. The presence of a cooling function makes the instability occur on a shorter timescale. We estimate critical Reynolds and magnetic Reynolds numbers of 40 and 20, respectively.

Interference Effects on Flow-Induced Oscillation of Three Tandem Cylinders With Passive Turbulence Control

Abstract The characteristics of flow-induced oscillation (FIO) of multiple tandem cylinders with passive turbulence control (PTC) change due to interference between PTC cylinders. Identifying positive and negative interferences is vital for designing marine structures involving PTC cylinders on elastic supports. Experiments on a single PTC cylinder, two PTC cylinders and three PTC cylinders in tandem were conducted to study the interferences due to placing an identical oscillating PTC cylinder upstream, downstream, and both upstream and downstream. Critical parameters (damping, stiffness) and Reynolds number (Re) are varied in the tests. The onset inflow velocity of oscillation is reduced and back-to-back vortex-induced vibration (VIV) and galloping is achieved, when placing interference cylinders in the vicinity of the basic oscillator. The interference due to the downstream cylinder enhances FIO of the basic oscillator. In the VIV region, one interference downstream cylinder is more effective than two interference downstream cylinders. In the galloping region, the enhancement improves for two interference cylinders for lower stiffness and damping ratios. The interference upstream cylinder tends to suppress FIO of the basic oscillator for higher stiffness. The suppression of one upstream cylinder is more vigorous than that of two upstream cylinders in the VIV region, and the opposite is true in the galloping region. When two interference cylinders are placed both downstream and upstream of the basic oscillator, its FIO is enhanced for the lower stiffness whilst it is suppressed for the higher stiffness.

Nonmodal stability analysis of Poiseuille flow through a porous medium

We unravel the nonmodal stability of a three-dimensional nonstratified Poiseuille flow in a saturated hyperporous medium constrained by impermeable rigid parallel plates. The primary objective is to broaden the scope of previous studies that conducted modal stability analysis for two-dimensional disturbances. Here, we explore both temporal and spatial transient disturbance energy growths for three-dimensional disturbances when the Reynolds number and porosity of the material are high, based on evolution equations with respect to time and space, respectively. Modal stability analysis reveals that the critical Reynolds number for the onset of shear mode instability increases as porosity increases. Moreover, the Darcy viscous drag term stabilizes shear mode instability, resulting in a delay in the transition from laminar flow to turbulence. In addition, it demonstrates the suppression of three-dimensional shear mode instability as the spanwise wavenumber increases, thereby confirming the statement of Squire’s theorem. By contrast, nonmodal stability analysis discloses that both temporal and spatial transient disturbance energy growths curtail as the effect of the Darcy viscous drag force intensifies. But their maximum values behave like O(Re2) for a fixed porous material, where Re is the Reynolds number. However, for different porous materials, the scalings for both temporal and spatial transient disturbance energy growths are different. Furthermore, increasing porosity also suppresses both temporal and spatial disturbance energy growths. Finally, we observe that temporal transient disturbance energy growth becomes larger for a spanwise perturbation, while spatial transient disturbance energy growth becomes larger for a steady perturbation when angular frequency vanishes. The initial disturbance that excites the largest temporal energy amplification generates two sets of alternating high-speed and low-speed elongated streaks in the streamwise direction.