Dynamic feedback control through wall suction in shear flows

Blood flow through a narrowed artery, such as those caused by plaque buildup, can lead to various cardiovascular diseases. Therefore, studying this phenomenon is essential for developing better treatments and understanding the underlying mechanisms. In this study, a mathematical model is developed to simulate blood flow through stenotic arteries with porous walls. The familiar Navier-Stokes equations are executed to simulate flow, whereas the Carreau fluid model is employed to quantify blood rheology. This model illustrates both Newtonian and non-Newtonian characteristics depending on the shear rate. Other than that, the blood vessel is assumed to have a moderate level of stenosis with porous saturated walls. The solution of the governed partial differential equations is carried out using a finite difference scheme. Our research addresses the impacts of permeability and rheological parameters, stenosis size, Brinkman number, and Prandtl number on blood velocity, flow rate, resistance to flow, wall shear stress, and temperature distribution. The findings demonstrate that while flow resistance reduces, the velocity, shear stress, and flow rate increase with high permeability values. Conversely, as the stenosis amplitude increases, the magnitude of wall shear stresses, velocity, and flow rates decrease. In addition, blood rheology also has the potential to transfigure the aforementioned factors. We also found that the temperature distribution increases with the Brinkman number, stenosis size, permeability, and power law exponent. It drives down, however, by raising the values of Weissenberg number and Prandtl number. Collectively, our findings contribute to a better understanding of the complicated phenomena of blood flow in stenotic vessels with porous saturated walls. The findings of this study may be valuable in establishing improved therapies for cardiovascular disorders and formulating more precise mathematical models to forecast blood flow in vivo.

Fouling layer growth and distribution at the interface of pressure-driven membranes

V OOI JEN and Hoogendoorn formulated an analysis of vapor flow in a heat pipe and built a laboratory model using a porous flat plate with injection and suction zones to represent the evaporator and condenser, respectively. Their paper is interesting and informative because it attacks a difficult problem from two viewpoints. The heat-pipe flow problem has attracted investigators for many years, but theoretical solutions have met with varying degrees of success. Few have attempted, as the authors have, to perform experiments to confirm their own analyses. The authors are to be commended for their care and attention to detail. Their paper, however, exhibits deficiencies in both the theoretical and laboratory models. One of the authors' objectives was to explore pressure as an independent variable for steady incompressible, twodimensional heat-pipe flow. Their claim to be the first to treat pressure as an independent variable is unfounded. Galowin et al. analyzed the flow in a cylindrical porous tube which represented the condenser section of a heat pipe. The axial static pressure and the suction velocity were permitted to vary according to conservation principles. A second-order, o^rdinary, nonlinear differential equation was derived and integrated numerically to obtain flow property distributions. In both analyses, constant radial and normal static pressures in the condenser zone and Poiseuille flow at the condenser inlet were assumed. A fundamental assumption used by van Ooijen and Hoogendoorn is that the suction velocity through the porous wall is constant in the condenser zone, viz., v(x,Q)=v0. Decoupling of the suction velocity in the flow equations is characteristic of similarity solutions. The analysis of Galowin et al. has revealed that flow distributions predicted by the similarity approach are incorrect. A lengthy explanation of separation zones in the condenser section for Rer > 10 is given. It is not clear whether recirculation is developed because of the numerical procedure or the physics of the flow. Many apparently suitable boundary conditions can show flow separation at stagnation points in sink or suction flow; however, many of these cases are not physically realistic. To prove their hypothesis the authors state that a sharp increase in measured pressure at the condenser end wall reveals a strong recirculation (in agreement with their numerical results). Actually, they have demonstrated only that the axial pressure increases in the vicinity of the stagnation point, which is in agreement with the analysis of Galowin et al. and the results of Quaile and Levy. Van Ooijen and Hoogendoorn emphasize that at higher values of Rer (e.g., 50) the stream patterns in the evaporation and condensation zones become strongly coupled. The conclusion is that any calculation denying this coupling will lead to incorrect results. The fact is that the flow has been decoupled a priori by assuming that v0 is constant no matter what Rer is chosen. A constant v0 can cause the appearance of recirculation zones to satisfy continuity and can account for the deviations from Poiseuille velocity profiles at high Rer. Thus, there appears to be a contradiction in their conclusion. The next part of our discussion concerns the laboratory model. The suction velocity cannot be uniform in their experiment because of the sintered steel plate. Studies conducted with sintered porous walls for rocket and jet engine cooling have demonstrated that both lateral and normal velocity components occur at the wall. More importantly, in this experimental arrangement most of the flow entering the condenser section will exit near the downstream end, because the only mechanism for changing the momentum vector is the stagnation zone and not the sintered porous plate. Only a very carefully designed thin porous wall containing small circular holes, i.e., orifices which vary in diameter and are distributed appropriately over the axial length, will produce a uniform outflow. The nonuniformity of v0 is borne out by Fig. 14 of the authors' paper which shows that the varying internal pressure levels are lower toward the downstream end of the condenser. The interpretation is that the flow velocity through the wall is higher where the larger pressure differences occur across the wall and variable over the length, not constant as they assume. In conclusion, we would like to offer two suggestions to improve an already promising study put forth by van Ooijen and Hoogendoorn. The analysis should be recast to permit variation of the suction velocity in the condenser section. This can be done by relating pressure to velocity in the form of Darcy's law:

A low Reynolds number flow in a channel obstructed by rows of ribs is of practical importance with respect to heat transfer enhancement. In order to obtain a high outlet gas temperature in a limited channel length, a flow rate should be low, whilst the flow should be turbulent to ensure an efficient heat transfer. Our preliminary study has indicated that the turbulence can be retained even at a low Reynolds number through the disturbance by the rows of the ribs in the channel. The ribs induce three-dimensional complex flow in their vicinity. The purpose of our research is to examine the influence of the ribs upon the flow and the temperature fields. Direct numerical simulation (DNS) of turbulent heat transfer in a fully developed channel flow has been performed at a friction Reynolds number of Reτ = 80 and a Prandtl number of Pr = 0.71, where Reτ is based on the friction velocity uτ0 , the channel half width δ and the kinematic viscosity ν. The friction velocity uτ0 is defined by the streamwise mean pressure gradient which drives the flow. The non-slip condition is adapted to the top and bottom walls. The periodic boundary condition is imposed in the horizontal directions. The rectangular rib is placed at the center of the computational domain (Fig. 1) and connected directly to both of the top and bottom walls. The rib is treated by the immersed boundary method. The constant wall temperature difference is applied for the temperature field. The flow rate decreases owing to the pressure drag of the rib, and the bulk Reynolds number Rem ( = um·2δ/ν ) drops down to 1120, where um is the bulk mean velocity. In the case of the plane channel without the rib, our additional calculation at this Rem has revealed that the flow becomes laminar. In the present case, the flow remains turbulent since the turbulent kinetic energy is produced in the vicinity of the rib. The instantaneous flow field is shown in Fig.1. Upstream of the rib, a horseshoe vortex is formed near the wall. The flow separates at the forward-edge of the rib. Downstream of the rib, a three-dimensional wake is formed, which contains a lot of vortices. Figure 2 shows the time-averaged local Nusselt number on the bottom wall. The heat transfer is significantly enhanced in front of the rib and the wake region. Upstream of the rib, the local Nusselt number is remarkably large. It indicates that the low temperature fluid in the central region is transported towards the bottom wall by the flow impingement induced by the rib. This region corresponds to that of the horseshoe vortex. In addition, Nu is large in the wake region as a consequence of effective mixing.

Couette–Poiseuille flow of Bingham fluids between two porous parallel plates with slip conditions

Computer simulations were used to investigate shear flows of large numbers of viscoelastic, monosized, spherical particles in unbounded and bounded systems with solid fractions ranging from 0.16 to 0.59. A modified hard-sphere model with inelastic, instantaneous particle interactions was found to replicate some results predicted by kinetic theory in an unbounded shear flow at low and moderate solids fractions. This model was found to predict features such as particle lateral diffusive motion even for systems at solid fractions as high as 0.56. However, for higher solid fractions where phenomena such as jamming could occur, a particle dynamics model accounting for particle contacts of finite duration has been developed, in which the viscoelastic behavior of the particles was represented using a nonlinear Hertzian model. The nonlinear viscoelastic model was found to give more reasonable predictions for cluster formation than previously reported linear models, especially when accounting for surface friction in the model. However, neither frictionless nor frictional particle models could predict particle ordering in unbounded flows. As such, simulations were performed for bounded systems using both the modified hard-sphere model and the nonlinear particle dynamic model. For a bounded shear flow, particle ordering could be predicted by the hard-sphere model even in the absence of both particle friction and gravity, with the local solid fraction and wall separation distance governing the flow stability. For these conditions chain formation was found to be quite likely in the disordered layers for frictional particles. The interesting stick-slip dynamics could be clearly observed in the normal stress signal at the bottom wall. Interpretations were proposed for the complex processes observed, which could lay the foundation for further investigations in sheared dense granular systems.

The acceleration of inertial particles in a homogeneous shear flow may vigorously respond to the intense flow structures induced by the mean shear. In this study, by direct numerical simulation (DNS) of particle-laden shear flow, we observe the statistical properties of those accelerations, and then we assessed the recently proposed simulation approach in which the effect of intermittency on residual scales is linked directly with coarsely resolved flow turbulence. First, we focused on the acceleration statistics of fluid particles in a homogeneous shear flow. Consistent with previous findings in homogenous isotropic turbulence, the norm and the direction of fluid acceleration in shear flow are shown to be conditioned by the dynamics of intermittent long-lived vortical structures. The averaged acceleration norm of the fluid particle exhibits a pseudo-cyclic behaviour, which is a signature of the periodic action of the largest confined vortices against the mean shear. The long correlation time of the acceleration norm, of the order of the integral time, the high level of fluctuations of the acceleration norm (very close to the magnitude of the mean norm of acceleration) and the log-normality in its statistical distribution reflect the impact of intense flow structures in shear flow. The presence of these zones results also in highly non-Gaussian statistics of the acceleration of a fluid particle and its velocity increments at small time lags. Contrary to the acceleration norm, the acceleration direction of a fluid particle is observed to be short, of the order of the Kolmogorov time, and to be statistically independent of the acceleration norm. The short-time correlation of the acceleration direction is attributed usually to effects of centripetal forces in intense vorticity filaments. We suggest that, in homogeneous shear turbulence, there may be a supplementary effect on the acceleration direction: the vortex, stretched by the mean shear, may exert the preferential direction of fluid particle acceleration. As evidence, our DNS shows that fluid particles are accelerated preferentially in the direction of longitudinal vortical tubes, effectively stretched by the imposed mean shear. Concerning simulations with heavy point-wise particles, it is shown that, when the inertia of a particle is not high, its acceleration closely follows all the aforementioned properties of the fluid particle acceleration. Particularly, an inertial particle is also entrained by accelerating motion in the direction of the effectively stretched vortical tubes. Although with increasing Stokes number of the inertial particle, the effects of strong intermittency in the flow are filtered, it is shown that the alignment of the particle acceleration direction with vortical tubes is amplified – particles with higher inertia respond to solicitations of stronger vortical structures. An alternative to the Maxey (J. Fluid Mech., vol. 174, 1987, pp. 441–465) preferential sweeping mechanism is discussed in this paper. When the shear turbulence is under-resolved, we employed the large-eddy simulation (LES) equations with a forcing term on the smallest resolved scales in order to simulate stochastically the effects of the dynamics on the residual scales. The forcing term is expressed with two independent stochastic processes, one for its norm and another for its direction. While the norm of acceleration is modelled using Pope’s log-normal process with the integral time for correlation, its direction is modelled in the framework of an Ornstein–Uhlenbeck process on the unit sphere. Consistently with our DNS, the latter process contains two presumed times: the homogeneous strain rate is specified as typical time of relaxation towards the direction of resolved vorticity, and the Kolmogorov time is presumed as a typical time of the diffusion process on the unit sphere. The high efficiency of this approach is demonstrated in prediction of the small-scale dynamics observed in DNS, even in the case when the shear length scale is not resolved by LES.

A direct numerical simulation is performed to examine the initial relaxation of spatially evolving turbulent channel flow subjected to uniform wall blowing and suction. Just beyond the entrance section, a uniform blowing is applied at the lower wall and a uniform suction at the upper. The relaxation processes of the mean velocity, turbulence intensities, Reynolds shear stresses, and vorticity fluctuations after the sudden wall blowing and suction are scrutinized. The effect of wall blowing and suction on the location and strength of the streamwise vortices is significant. The relaxation associated with suction is much slower than that with blowing. The initial relaxation is evaluated in terms of the turbulent kinetic energy transport. Blowing enhances the intercomponent energy transfer between the Reynolds stresses, whereas suction suppresses it.

The effectiveness of the opposition control method proposed by Choi et al. [J. Fluid Mech. 262, 75 (1994)] has been studied using direct numerical simulations. In this study, the effects of the amplitude and the phase of wall blowing and suction control input were considered separately. It is found that the amplitude of wall blowing and suction as well as the detection plane location played an important role in active control for skin-friction drag reduction. By changing the amplitude, a substantial drag reduction was achieved for all detection plane locations considered, and the efficiency of the opposition control was also improved. When the control was effective, the drag reduction was proportional to the wall blowing and suction strength. There existed a maximum wall blowing and suction strength, beyond which the opposition control became less effective or even unstable. Turbulence characteristics affected by various wall blowing and suction parameters were analyzed to understand the underlying mechanisms for drag reduction. The wall-normal velocity and vorticity fluctuations showed a strong correlation with drag reduction.

Dynamic feedback control of day-to-day traffic disequilibrium process

An incompressible, pressure‐driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress with the first and the second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary‐value problem for a second‐order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. The case of near‐critical transpiration, when the shear stress at the injection wall vanishes, is considered. It is shown that the maximum point on the mean velocity profile lies in a thin sublayer near the suction wall in this case. A formula for the position of the maximum point as a function of the transpiration factor is obtained. The mean velocity profiles near the suction wall are calculated. A friction law for Poiseuille flow with near‐critical transpiration is found, which makes it possible to describe the relation between the shear stress at the wall, the Reynolds number, and the transpiration velocity by a single function of one variable. Direct numerical simulation of the flow for some transpiration factors is performed. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Solutions of the interfacial shear and normal stresses in plate flexural-strengthened beams based on different complementary strain energy assumptions

Direct numerical simulations are performed to study temporal variations of the wall shear stresses and flow dynamics in the turbulent pulsatile pipe flow. The mechanisms, responsible for the paradoxical phenomenon for which the amplitude of the oscillating wall shear stress in the turbulent flow is smaller than that in the laminar flow for the same pulsation conditions, are investigated. It is shown that the delayed response of turbulence in the buffer layer generates a large magnitude of the radial gradient of the Reynolds shear stress near the wall, which counteracts the effect of the oscillating pressure gradient on the change of the streamwise velocity and hence reduces the amplitude of the wall shear stress. Such a delayed response consists of two processes: the delayed development of near-wall streaks and the subsequent energy redistribution from the streamwise velocity fluctuation to the other two co-existing components. This is a dynamical manifestation of the viscoelasticity of turbulent eddies. As the frequency is reduced, the variation of the friction Reynolds number results in a phase-wise variation of the time scale and intensity of the turbulence response, causing the hysteresis of the wall shear stress. Such a phase asymmetry is amplified by the increase of the pulsation amplitude. An examination of the energy spectra reveals that the near-wall streaks are stretched in the streamwise direction during the acceleration phase, and then break up into small-scale structures in the deceleration phase, accompanied by the enhanced dissipation that transforms the turbulent kinetic energy into heat.

Direct numerical simulation (DNS) is performed for two wall-bounded flow configurations: laminar Couette flow at $Re=740$ and turbulent channel flow at $Re_{\unicode[STIX]{x1D70F}}=180$, where $\unicode[STIX]{x1D70F}$ is the shear stress at the wall. The top wall is smooth and the bottom wall is a realistically rough superhydrophobic surface (SHS), generated from a three-dimensional surface profile measurement. The air–water interface, which is assumed to be flat, is simulated using the volume-of-fluid (VOF) approach. The two flow cases are studied with varying interface heights $h$ to understand its effect on slip and drag reduction ($DR$). For the laminar Couette flow case, the presence of the surface roughness is felt up to $40\,\%$ of the channel height in the wall-normal direction. Nonlinear dependence of $DR$ on $h$ is observed with three distinct regions. A nonlinear curve fit is obtained for gas fraction $\unicode[STIX]{x1D719}_{g}$ as a function of $h$, where $\unicode[STIX]{x1D719}_{g}$ determines the amount of slip area exposed to the flow. A power law fit is obtained from the data for the effective slip length as a function of $\unicode[STIX]{x1D719}_{g}$ and is compared to those derived for structured geometry. For the turbulent channel flow, statistics of the flow field are compared to that of a smooth wall to understand the effects of roughness and $h$. Four cases are simulated ranging from fully wetted to fully covered and two intermediate regions in between. Scaling laws for slip length, slip velocity, roughness function and $DR$ are obtained for different penetration depths and are compared to past work for structured geometry. $DR$ is shown to depend on a competing effect between slip velocity and turbulent losses due to the Reynolds shear stress contribution. Presence of trapped air in the cavities significantly alters near-wall flow physics where we examine near-wall structures and propose a physical mechanism for their behaviour. The fully wetted roughness increases the peak value of turbulent intensities, whereas the presence of the interface suppresses them. The pressure fluctuations have competing contributions between turbulent pressure fluctuations and stagnation due to asperities, the near-wall structure is altered and breaks down with increasing slip. Overall, there exists a competing effect between the interface and the asperities, the interface suppresses turbulence whereas the asperities enhance them. The present work demonstrates DNS over a realistic multiphase SHS for the first time, to the best of our knowledge.

Pore-scale insight into the effect of porous walls on fluid flow in rough-walled fractures