Effect of gap height and non-linearity of incident Couette-Poiseuille flow on aerodynamic characteristics of a square cylinder near wall

*Rajesh Bhatt1), Dilip K. Maiti2) and Md. Mahbub Alam3)#

1&3)Institute for Turbulence-Noise-Vibration Interaction and Control, Shenzhen

Graduate School, Harbin Institute of Technology, Shenzhen, China 2) Department of Applied Mathematics with Oceanology and Computer Programming,

Vidyasagar University, Midnapur 721102, India #alamm28@yahoo.com

ABSTRACT

A numerical study on the Couette-Poiseuille based non-uniform flow over a square cylinder of height a* placed parallel to the wall at different gap ratios (L=H*/a*: 0.1≤L≤2.0, where H* is the gap between the cylinder and wall) is conducted for different inlet non-dimensional pressure gradients P. The governing equations are solved numerically through finite volume method based on SIMPLE algorithm on a staggered grid system. Both P and L have an appreciable effect on the vortex shedding frequency, aerodynamic forces and the flow structure around the cylinder. At L≤0.25 for P≤3 the flow remains steady similar to the linear shear flow (P = 0). While increasing the value at P=5, unsteadiness developed in the far field propagates towards the cylinder, becoming quasi-steady in nature for L = 0.1. A further increase in L results in stretching of the shear layers from the cylinder; for a high P the flow deflects towards the downstream side with irregular transverse and longitudinal vortices. Collecting all the simulated results obtained, four distinct flow patterns I, II, III and IV are identified based on the vorticity contours, which are steady flow, unsteady single row vortex, two stretched row of vortices and trailing edge separation with complex wake, respectively. The critical gap height for onset of the vortex shedding decreases with increasing the value of P. The intensity of peak in the power spectra of the lift coefficient increases with the increase in P. At large L, the sharp peak changes into broad band and the wake interaction becomes complicated depending on P and L. An attempt is made to find a universal shedding frequency that is independent of L for different nonlinear velocity profiles. NOMENCLATURE a* :height of the cylinder (m) L :nondimensional gap height from cylinder to plane wall P :nondimensional pressure gradient Re :Reynolds number (Ua*/ν) CD :time averaged drag coefficient CL :rms of lift coefficient St :Strouhal number U :velocity at distance h* from the wall (m/s) V : kinematic viscosity of fluid

1. INTRODUCTION In many practical cases the flow approaching the structures, for example offshore platforms, is not uniform because of the velocity gradients from the incoming flow. Although high Reynolds number flows appear in most practical examples, the flow at low Reynolds number has an application in small heat exchangers and electronic cooling components. The difference between the flow velocities at upper and bottom part of the structure in the wall proximity makes flow somewhat more complex, resulting in different wake flow patterns. Therefore, it is very important to understand the basic flow phenomena from a cylinder under the influence of a non-uniform velocity profile near the wall. This study investigates the effect of the gap height L(=H*/a*, where H is the gap between wall and the cylinder and a* is height of the cylinder) and non-linearity of approaching velocity profile based on inlet pressure gradient P on flow characteristics of a square cylinder. Numerical studies on flow past a square cylinder for different inlet shear parameter K and at different Reynolds numbers Re were conducted by Hwang et al. (1997, Re=500-1500, K=0-0.25), Cheng et al. (2005, Re=100, K=0-0.5), Cheng et al. (2007, Re=50-200, K=0-0.5), Lankadasu and Vengadesan (2008, Re<100, K=0-0.2). The results showed that the Strouhal number (St), mean drag (CD) and fluctuating forces decreases with increasing K. Cheng et al. (2007) conducted simulation at Re=100, 0≤K≤0.5 for flow over a square cylinder and concluded that unlike that for a circular cylinder, the shedding frequency for the square cylinder decreases with increasing K. Lankadasu and Vengadesan (2008) reported that with increasing K, the critical Re, at which flow becomes unsteady, is reduced. Similarly several numerical studies have been conducted on rectangular cylinder of different aspect ratios under uniform and shear flow, e.g. by Sohankar (2008, Re=105), Islam et al. (2012, Re=100-250), Yu et al. (2013, Re=105) and Cao et al.(2014, Re=22000). These studies have shown that Re effect is less significant compared to other parameters (e.g., incoming velocity profile). Cao et al. (2014) concluded that the linear movement of the stagnation point to the high velocity side with increasing K is an inherent behavior in shear flow. Several studies on effect of wall proximity of a cylinder have also been carried out (Bailey et al., 2002; Lee et al. 2005; Kumar and Vengadesan, 2007; Wang and Tan, 2008; Dhinakaran, 2011; Maiti, 2012; Samani and Bergstorm, 2015). Bailey et al. (2002) conducted an experimental study and examined the vortex shedding from a square cylinder near a wall at Re=19000 and their results showed that the suppression of vortex shedding occurs for L<0.4. Lee et al. (2005) studied effect of aspect ratio and L(0.3≤L≤2) and introduced simple passive control method to reduce the aerodynamic drag and vortex-induced oscillation. They observed that the vortex begins to shed by the interaction of the separated shear layer on the upper surface of the cylinder with the upwash flow from the gap region. Wang and Tan (2008) experimentally compared the near-wake flow patterns for a circular and a square cylinder for different L ranging from 0.1 to 1.0. They concluded that the vortex shedding strength in the case of square cylinder is relatively weaker, as compared to that of the circular cylinder at the same L, and the critical L for vortex shedding are 0.3 and 0.5 for the circular and square

cylinders, respectively. Maiti (2012) reported that aerodynamic characteristics of a square cylinder under uniform shear flow are more sensitive to L for L>1 than for L ≤1.0. Dhinkaran (2011) observed an increment in aerodynamic forces with a decrement in L for the flow past a square cylinder placed near a moving wall at Re=100. The flow behavior was classified into two-row vortex street (1≤L≤4), single-row vortex street (0.4≤L≤1), quasi-steady vortex street (L=0.3) and vortex shedding suppression (L<0.3). Recently, Samani and Bergstrom (2015) conducted large eddy simulation to examine the wall proximity effect of a square cylinder for three values of L=0, 0.5 and 1.0 at Re=500. An increase in L leads to an increase in mean lift force (CL) and a decrease in mean drag (CD) coefficients. As the cylinder approaches the wall, the wake flow becomes increasingly asymmetric with the top recirculation cell displaced significantly upward and further downstream, and a secondary recirculation zone develops on the bottom wall. The aforementioned studies on the flow around a single cylinder are based on either uniform flow or linear shear flow with and without wall proximity. It is clear from these studies that the inlet velocity profile and the gap flow greatly altered the aerodynamic forces and flow around the cylinder. However, numerical studies on a square cylinder in wall proximity under non-uniform velocity profile are scarce. Therefore, the present study is aimed at understanding the effect of inlet non-linear velocity profile on aerodynamic forces and the wake flow structure of a square cylinder at different L. 2. PROBLEM DESCRIPTION With a wall lying along the x*-axis, a cylinder of square cross section of height a* is placed parallel to the wall at gap height H* from the wall (Fig.1(a)). The inflow and outlet boundaries lie at a distance 10a* and 20a* from the front and rear face of the cylinder, respectively. The top lateral boundary lies at 10a* from the plan wall. The upstream flow field is taken as Couette-Poiseuille flow based nonlinear velocity profile u*(y*). This upstream condition is consistent with the Navier-Stokes equation and viscous effects, because of no-slip requirement at the wall (Schlichting, 2000). In other words, the cylinder is sub-merged into the boundary layer of a plane wall. The following nonlinear velocity profile (with characteristic velocity U at y* = 10a*) is considered at the inlet:

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The nondimensional quantities V=(u, v), and p and t denote the velocity, pressure and time, respectively, with characteristic length and velocity are a* and U, respectively. No-slip condition is used on the cylinder surfaces and on the plane wall. A Dirichlet boundary (u=u(y), v=0) is used at the inlet boundary, while the Sommerfeld condition (∂ɸ/∂t)+uc(∂ɸ/∂x)=0, where ɸ is any flow variable, and uc is the local wave speed) is applied at the outlet. A slip boundary condition (∂u/∂y=0, v=0) is imposed on the top lateral boundary. The numerical treatment to the Sommerfield condition has been discussed in details in the previous study (Maiti, 2011). A finite volume method on a staggered grid system and then the pressure correction based iterative algorithm SIMPLE (Patankar, 1980) are applied. A third-order accurate QUICK (Leonard 1995) is employed to discretize the convective terms and central differencing scheme for diffusion terms. A fully implicit second-order scheme is incorporated to discretize the time derivatives. At the initial stage of motion, times step is taken to be 0.0001 which has been subsequently increased to 0.001 after the transient state. 3. GRID INDEPENDENCE AND VALIDATION OF CODE In this study, a nonuniform grid distribution is considered in the domain, with a uniform grid along the cylinder surface and increasingly enlarged grids away from the cylinder surface. The grid distribution in x-and y-directions is shown in Fig.1(c). The grid distribution in x - and y - directions is similar to that used in our previous study, Maiti (2012). Depending on the size of the distributed grids, the horizontal and vertical lines of the computational domain are divided into five and three, respectively, distinct segments. Along the y-direction, a uniform grid with size 0.005 is considered between wall and top of the cylinder (within the segment ‘Iy’ in Fig.1(c)). With reference to Franke et al. (1990) and Sohankar et al. (1997), the value of the first grid size from the cylinder is kept constant as 0.004 for the present computation. The grid refinement study on the number of uniform nodes and on the resolution of grid near the cylinder surface and the far-fields was conducted in the previous study (Maiti, 2011, 2012 and Maiti and Bhatt, 2015) for the case of single and inline tandem cylinders. A detailed discussion on the percentage of deviation has been made in these studies. In the present study the grid distribution along the y-direction varied according to the gap height of the cylinder from the wall. The numerical code validated for the case of a square cylinder placed near a wall (Bhattacharyya et al., 2006) has been used in the present study. Thus, the previous published results by the authors also show the validity of the used code for this work. In the present study the following operational dimensionless parameters affecting the flow are considered:

L= 0.1, 0.25, 0.5, 1.0, 1.5 and 2.0 Inlet pressure gradient P = 0, 1, 3, 5 Reynolds number (Re): 1000 (based on velocity at height 10a* from wall (Maiti

and Bhatt (2015))). Davis et al. (1984) found a good agreement between the numerical and experimental results for Re around 1000 for uniform flow.

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averaged d

oefficient (pattern I a

ow for P=0 ributes eithP. Quick inr all L an

er. At L=2 5 case com

with the incet al. (199e coefficieder. Maiti ase in L fo

ency and s

ts of the cnd L = 0.25power specFor P = 0, th increasinults in a c

multiples n of the wa

ncreases m

drag coeffi

(CL) in L-Pand II), andcase com

her supprencreasing nd P. Hig

the fluctumpared tocrease in L99) and Dients is asso

(2012) obor the unifo

spectra

cylinder as5, P ≤ 3 anctra. For hione single

ng L. For hcomplicatepeaks cor

ake with wa

monotonica

cients (CD)

P plane ald quickly inpared to th

ession of vregime is her fluctuuation of t P=0. As

L the stagnpankar anociated wit

bserved theorm shear f

functions nd L = 0.5igher valuee dominanhigher valued wake (rrespond toall vortices

ally with P a

) and (b) rm

lso consistncreasing he higher vvortex shed

corresponation at hthe lift coeit can be s

nation poind Sengupth the move similar trflow.

of L and P, P = 0, thes of L andnt peak is iues of P anFig. 3) ano the cylin

s.

at a particu

ms lift coef

ts of two regime (fo

values of Pdding or vends to the higher P efficient eseen from

nt is shiftedpta (2005) vement of rend of inc

P are prese flow wasd P, a definidentified (nd L, the wnd multiplender own s

ular L.

fficient

regimes: or pattern P. Similar ery small onset of indicates nhanced the flow

d into the reported the front

crease in

sented in s steady, nite peak (Fig. 8a), wake and e and/or shedding

F

Fig. 9 prcylinder considerplaced increaseis contra

Fig. 8. The

Fig.

resents thefor differe

red as the in nonuni

es when P ary to the

power spe

9. Contour

e contour ent L and St. As demform flow

P is further finding of

ectra of flu

rs of Strou

plots of StP. The do

monstratedcrucially

increasedSaha et a

uctuating lif

hal numbe

trouhal numominant ped in Figs. 2depends from 0 to

al. (1999)

ft coefficien

er (St) at di

mber (St) oeak in Fas2 & 3, the w

on P an5. The incand Lanka

nt at differe

fferent L a

of the vortst Fourier wake topold L. The

crease in Sadasu & V

ent P and L

and P.

tex sheddintransform logy of the

Strouhal St for preseVengadesa

L.

ng of the (FFT) is

e cylinder number

ent study n (2008)

who reported that St either remain constant or decreases with increase in the incoming shear rates. The maximum St (=0.18) is identified in pattern-II for L=0.25, P=5, where the wake consists of a single row of regular vortices. St of the dominant peak is sensitive to P. In general for pattern-III and IV, St decreases with increase in L for P ≥ 1. At large L, the effective Reynolds number increases and vortices are stretched, it results smaller value of St. These observations are consistent with Maiti (2012) who reported drop in St at higher L for a uniform shear flow past square cylinder.

It has been observed here that significant changes occur in St with change in L particularly for higher values of P. For P≤3 variation in St is less with variation in L, compared to the higher value of P ≥ 3. The variation in St is corresponding to change in wake flow pattern. Roshko (1954) define a universal Strouhal number (St*) based on the hypothesis that the wake of different bluff bodies are similar in structure, as

)5(*

*

a

w

u

USt

u

wfSt

bb

Where f is the shedding frequency, w is the lateral distance between the two free shear layers and is the time mean velocity at the boundary layer separation defined as

)6(1 P

bC

U

u

Where CP is the base pressure. The concept of a universal Strouhal number has been used by many researchers with different definition of wake widths (Griffin & Ramberg 1974, Roshko 1954, Alam & Zhou 2007 and Alam et al. 2011). All these studies are based on higher Reynolds number uniform flow. So far, nobody has used this theory for the single cylinder with nonuniform inlet flow in wall proximity. An interesting question arises is : for a nonlinear inlet flow, is it possible to find the universal Strouhal number that is independent of L ? It is clear from the flow structures that the presence of incoming nonlinear flow and the wall confinement produce an asymmetry in the two separated shear layers emerging from the upper and lower surfaces of the cylinder. Therefore, it is difficult to measure the wake width for the present case. For the St* calculation we have considered cylinder height (a*) as the characteristic length instead

of the wake width w. The values of St* are calculated as, buStSt /* and plotted

along with St in Fig. 10. It is clear from Fig.10, St and St* curves are qualitatively similar. For linear shear flow case (P=0), the values of St* collapsed to fairly constant value with less variation (0.037±0.003) with L. However, there is large variation in St* for P=5 (St*=0.085 ± 0.03). The higher variation in St* for larger P can be explained based on that increase in P is similar to increase in Re. Richter and Naudascher (1976) observed variation in St* (0.1560-0.2800) with change in Re from 2×104 -1.0×105 to 2×105-4×105. Here also for P≥3, we observe large variation in St* ≈ 0.03 - 0.1 for P=3 and St* ≈ 0.05-0.13, for P = 5 with L. However, the range of variation in St* is (0.036 - 0.04) & (0.04 - 0.058) for P = 0 & 1, respectively. Large variation at lower L(≤1.0) is attributed the wall proximity effect, which insinuates that the defining of universal Strouhal number St*

does noFig. 10 tthat RosP for L≥

Fig. 10. for differ 5. CONC The effea squargradientflow strubased oP=0, L=it conveincomingin patterand two is obserchange large L becomepower sdependsfor P ≥

ot fit for thethat for L≥shko theor1.0.

Variationsrent P.

CLUSIONS

ect of gap hre cylindert (P) and gucture, forcon the cont0.5 in patte

erts into cog flow at Prn-II. In parow of vor

rved from tin either Pand P in s complic

spectra. Ths on P. The3 compar

e flow past1.0, variat

ry for findin

s in Strouh

S

height andr has beegap height ces and shtours of spern-I. The

omplete voP=5. Unstettern-III shrtices stretthe cylinde

P or L whicpattern-IV,ated, whic

he critical ge sheddingred to P=0

t square cyion in St* ing St* bas

hal numbe

d inlet-flow en investig

(L) have hedding frepanwise vounsteadine

ortex sheddeady periodhear layerstched in theer for all thch play a m, trailing ech resultsgap heightg frequenc0 case. Th

ylinder in wis less par

sed on

r (St) and

nonlinearigated numan apprec

equency. Forticity. Thess is geneding at L=dic flow wi

s from the e streamw

he cases. Tmeasure rodge separStrouhal

t for onset cy of the cyhe St* va

wall proximrticularly foprovides b

modified S

ty on the cmerically. Wciable effecFour distince flow is serated for

=0.25 by inith a singlebottom of

wise directioThe gap floole in the foration occupeak becor suppre

ylinder decriation sho

mity. As it or small vabetter appr

Strouhal n

characterisWe found ct on the dct flow pattsteady for L=0.1 at fa

ntroducing e row of vothe cylindeon. For L ≥ow changeormation ours and thcomes brossion of th

creases witows that R

is evident alues of P. roximation

number (St

stics of flowthat the

dependencterns are cP≤ 3, L≤ 0

ar downstrnonlineari

ortices is oer remain ≥ 1.0 unstees appreciaof wake voe wake in

oad bandehe vortex sth increaseRoshko’s t

from the It shows at lower

t*) with L

w around pressure

ce on the classified 0.25 and eam and ity in the observed attached

eady flow ably with rtices. At teraction d in the shedding e in the L theory of

universal Strouahl number does not fit for higher P≥3 and lower L≤1.0. The values of CD and CL are insensitive to change in L for P=0 and upto L=0.5 for P ≥ 1. For L>0.5, CD increases almost linearly with L. The average pressure coefficient PC shows significant change along the surface of the cylinder with increase in L and P that result in an increase in the value of CD. ACKNOWLEDGMENTS Alam wishes to acknowledge the support given to them from National Natural Science Foundation of China through Grants 11672096 and from Research Grant Council of Shenzhen Government through grant JCYJ20160531191442288. REFERENCES Alam, M.M. and Zhou, Y. (2007), “Turbulent wake of an inclined cylinder with water running”, J. Fluid Mech. 589, 261-303. Alam, M.M., Zhou, Y. and Wang, X.W., (2011), “The wake of two side-by-side square cylinders”, J. Fluid Mech. 669, 432-471. Bhattacharyya, S. and Maiti, D.K. (2004), “Shear flow past a square cylinder near a wall”, Int. J. Eng. Sci., 42, 2119-2134. Bhattacharyya, S., Maiti, D.K. and Dhinakaran, S., 2006, “Influence of buoyancy on vortex shedding and heat transfer from a square cylinder in wall proximity”, Numer. Heat Transfer, Part A, 50(6), 585–606. Bailey,S.C.C., Kopp, G.A. and Martinuzzi, R.J. (2002), “Vortex shedding from a square cylinder near a wall”, J. Turbulence, 3, 003. Cao, S., Zhou, Q. and Zhou, Z. (2014), “Velocity shear flow over rectangular cylinders with different side ratios”, Computers & Fluids, 96, 35-46. Cheng, M., Whyte, D.S. and Lou, J. (2007), “Numerical simulation of flow around a square cylinder in uniform-shear flow”, J. Fluids Struct., 23, 207-226. Cheng, M., Tan, S.H.N. and Hung, K.C. (2005), “Linear shear flow over a square cylinder at low Reynolds number”, Physics of fluids, 17, 078103. Dhinakaran, S. (2011), “Heat transport from a bluff body near a moving wall at Re= 100”, Int. J. Heat Mass Transfer, 54, 5444-5458. Dipankar, A. and Sengupta, T.K. (2005), “Flow past a circular cylinder in the vicinity of a plane wall”, J. Fluids Struct., 20, 403–423. Davis, R.W., Moore, E.F. and Purtell, L.P. (1984), “A numerical-experimental study of confined flow around rectangular cylinders”, Phys. Fluids, 27, 46–59. Franke, R., Rodi, W. and Schonung, B., (1990), “Numerical calculation of laminar vortex shedding flow past cylinders”, J. Wind Eng. Aerodynamics,35(1–3), 237– 257. Griffin, O.M. and Ramberg, S.E. (1974), “The vortex street wakes of a vibrating cylinders”, J. Fluid Mech. 66, 729 -738. Hwang, R.R. and Sue, Y.C. (1997), “Numerical simulation of shear effect on vortex shedding behind a square cylinder”, Int. J. Num. Meth. Fluids, 25, 1409-1420. Islam, S.U., Zhou, C.Y., Shah, A. and Xie, P. (2012), “Numerical simulation of flow past rectangular cylinders with different aspect ratios using the incompressible lattice

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