Diffusion in inhomogeneous flows Unique (1)

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Diffusion in inhomogeneous flows: Unique equilibrium state

in an internal flow

Tapan K. Sengupta

⇑

_{, Himanshu Singh, Swagata Bhaumik, Rajarshi R. Chowdhury}

Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India

a r t i c l e

i n f o

Article history:

Received 2 November 2012 Received in revised form 10 July 2013 Accepted 2 October 2013

Available online 12 October 2013

Keywords:

Navier–Stokes equation Inhomogeneous flow Enstrophy transport equation Direct numerical simulation Diffusion

Rectangular lid driven cavity

a b s t r a c t

The role of diffusion in creating rotationality (enstrophy) is studied here and a transport equation for enstrophy is derived to explain this connection. As an illustration, flow instabilities and pattern formation are investigated here for an inhomogeneous internal flow with definitive boundary conditions. Results obtained by direct numerical simulation (DNS) of flow inside a two-dimensional rectangular lid driven cavity (RLDC) show that diffusion is responsible in forming patterns at a post-critical Reynolds numbers. The transport equation for enstrophy derived from the Navier–Stokes equation in Eulerian framework helps to explain the enstrophy spectrum in flows, specially in 2D flows, where vortex stretching is absent as the dominant energy cascade mechanism to small scales. For the 2D flow in RLDC, diffusion and convection provide a unique equilibrium state in an intermediate post-critical range of Reynolds number around 6000. This is independent of the geometric aspect ratio (height to width of the cavity) of the cavity greater than or equal to two. Such equilibrium can be observed in numerical simulations, only when special care is exercised for diffusion discretization at high wavenumbers. Another motivation in this work is to show that diffusion and dissipation are not identical for inhomogeneous flows, as opposed to equating these in studies of homogeneous turbulent flows. Organized enstrophy is shown as a consequence of over-riding action of diffusion in creating rotationality in this flow.

1. Introduction

Investigation on the true role of diffusion has remained a problem, ever since the time when its role was considered as stabilizing fluid flow by damping disturbances, attributed to Kelvin, Helmholtz and Rayleigh[1]. Equating viscous diffusion with dissipation was the sole reason for early instability studies to ignore diffusion, as discussed in[1,2]. However, such studies were unable to explain instability of flow over a flat plate, while the same flow was successfully investigated by solving Orr-Sommer-field equation (OSE) [3–5], which includes viscous diffusion in the formulation. It was thought that retaining diffusion is equivalent to producing an appropriate phase shift for a positive feedback, which leads to flow instability.

Doering and Gibbon [6] studied the enstrophy transport for two-dimensional periodic flows and obtained the evolution of integrated enstrophy over the full domain as

d dt

1 2k

x

k

2 2

¼

m

k

r

x

k22 ð1Þ

where

x

is the vorticity and

m

is the kinematic viscosity. Here, the enstrophy is defined over the full periodic domain byk

x

k2₂. Thus, one notes the effects of diffusion as strictly dissipative for periodic flows viewed globally. In performing DNS of flows, one discretizes all the terms and obtains the numerical solution without any ambiguity. However, the point of view of equating diffusion with dissipation is often used, as given above in Eq. (1), while interpreting DNS results of homogeneous turbulent flows [7]. However if diffusion is viewed instantaneously at any point in a flow, then the effects of diffusion is not strictly dissipative, as will be explained here. When one looks at the time-averaged kinetic energy of turbulent flows globally, effects of diffusion is again seen to be as dissipative[8,9]. As shown in Eq. (4.34) of[9], time-average of the diffusion term of Navier–Stokes equation manifests itself as a combination of (i) a strictly dissipation term and (ii) anotherviscous transferterm. However,the viscous transfer term integrates to zero over the whole flow by the divergence theorem. This term is sometimes also referred to as diffusive, because it is zero for homogeneous turbulence. The authors furthermore add that the viscous transfer term is negligible at high Reynolds numbers, except within the thin viscous layers very near any solid surfaceswhileon the other hand, the dissipative term is of crucial importance to turbulence energetics everywhere. Similar observations are made in Section 3.3 of[8], with respect to time-averaged turbulent kinetic energy. In the present

⇑ Corresponding author. Tel.: +91 512 2597945. E-mail address:[email protected](T.K. Sengupta).

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investigation, we look at the instantaneous local behavior of diffu-sion term and demonstrate in a flow the existence of a unique equi-librium state where the diffusive nature dominates over the dissipative nature of the viscous term in Navier–Stokes equation.

We also show by an appropriate analysis requiring the consid-eration of the time-accurate total mechanical energy (as opposed to time-averaged property of only the kinetic energy) of the flow for which the action of viscous terms is not directly apparent. If one constructs an equation for the total mechanical energy, as sug-gested in[10]and developed in[11], then the role of diffusion be-comes clearer, as described in the following. One writes the Navier–Stokes equation in rotational form for this analysis as

@~V

@t~V~

x

¼

r

p

q

þ

~ V~V

2 !

þ

m

r

x

~ ð2Þ

where different variables represent their usual meanings and the viscous effects is via the last term on the right hand side, written as the curl of the vorticity vector, multiplied by the kinematic viscosity.

Describing the total mechanical energy (E) by

E¼p

q

þ

~ V~V

2

and taking a divergence of the above Navier–Stokes equation yields the distribution ofEby the following equation

r

2E¼

r

ð~V

x

~Þ ð3Þ

Note that the viscous term drops out identically due to a vector identity and the right hand side originate strictly from convection term. However, the right hand side of the above equation can be ex-pressed using the vector identity in further simplification of this equation

r

ð~V

x

~Þ ¼

x

~

x

~~V ð

r

x

Þ

Denoting the instantaneous point property of enstrophy by X1¼

x

~

x, Eq.

~ (3)can be written as

r

2E¼

X

1~V ð

r

~

x

Þ ð4Þ

This equation shows the relevance of enstrophy and the diffusion operator to be central in distributing total mechanical energy. In

[7], a similar equation has been written for the static pressure (see Eq. (1.2) of the reference) which in present notations is given by

r

2 p

q

¼ ð

X

1

=

m

Þ=2 ð5Þ

where

= 2msijsijandsijis the symmetric part of the strain tensor.

This equation is wrongly stated to be valid only for homogeneous turbulence. Eq. (4) is written for any general flow derived from Navier–Stokes equation without making any assumption or simpli-fication. One notes that the term on the right hand side of Eq.(4)

can be written as~_Vr2~V

m , in drawing an analogy with the term

/m,

on the right hand side of Eq.(5), even though the right hand side of Eq. (4)purely originates from convection term. This source of confusion prompted the authors in [7,12,13], to equate the roles of enstrophy and dissipation. One of the motivations here is to highlight the connection between diffusion and enstrophy for flows. The development and use of total mechanical energy equation to study any flow instability is described in detail in[1,11].

In trying to understand the role of diffusion in creating rotation-ality, an evolution equation is also developed here for enstrophy, as a point property and its higher powers for any flow. This exercise explains the roles of diffusion, dissipation and creation of rotation-ality progressively to smaller scales. To demonstrate that this is

valid for any flow, we focus on a 2D flow, which does not have the presence of vortex stretching to create smaller scales.

Reported DNS in [7], used Fourier spectral discretization in space and second order Runge–Kutta time integration to solve Na-vier–Stokes equation. This space–time dependent discretization is very restrictive in parameter space, due to its numerical instability and also due to its high dispersion error, as shown by spectral anal-ysis in the appendix using the 1D convection equation. It is obvious that any method which cannot solve this simple convection equa-tion, is practically of little use in solving more complex Navier– Stokes equation. The dynamical equilibrium in flows is a balance between convection and diffusion processes, both of which have to be captured correctly in equal measure. One of the salient fea-tures of the presented results here is to show the existence of a universal equilibrium between convection and diffusion in Na-vier–Stokes equation. This can be captured only by carefully de-signed numerical methods explained in the next section and appendix.

There have been significant progresses made in developing high accuracy compact schemes, which are dispersion relation preserv-ing (DRP) and has been used for inhomogeneous flows. A similar method has been used in[14,15]to simulate an inhomogeneous zero pressure gradient boundary layer from the receptivity to a fully developed 2D turbulent stage, displaying k3 spectrum for the energy. One of the motivations here is to show that for 2D flows, rotationality is created at different scales via the enstrophy cascade. This establishes a link between diffusion and enstrophy for a wall-bounded inhomogeneous flow.

Here, the flow inside a RLDC driven by uniform translation of the top lid (U1) is used as an example to reveal the role of diffusion

in Navier–Stokes equation, where pronounced rotationality is cre-ated by simple translation of the top lid. It is well known[16]that turbulence is characterized by many attributes, out of which the primary ones being rotationality and broad-band energy spectrum created by various instability mechanisms.

Flow in a square LDC has been studied and a unique topology (triangular core vortex and gyrating satellite vortices) is described in[17,18]. This was obtained with the help of highly accurate dis-cretization of convection and diffusion processes in the flow. Flow in RLDC is more complicated due to the presence of multiple cells having distinct vortical structures. The upper cell of RLDC resem-bles the flow in a square LDC, which in turn, drives the cell below and so on. The rotational flow structures seen in various cells of RLDC are caused by the translational motion of the lid, with each cell showing presence of vortices of both signs.

The manuscript is formatted in the following manner. In the next section, governing equations and the numerical methods to solve 2D flow inside the RLDC are described. This is followed by a section describing the flow inside RLDC, with respect to the insta-bility sequence, topology and Hopf bifurcation of the flow. To ex-plain this instability sequence and induced rotationality, transport equation for enstrophy has been derived in Section 4. In Section5, we emphasize the requirements on diffusion discret-ization in DNS. This is followed by summary and conclusion of the results. In the appendix, the spectral analysis of numerical schemes used for convection equation has been carried out.

2. Governing equations and numerical formulation

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@

x

where

x

is the non-zero component of vorticity in thez-direction and~_V _{is the velocity and}_Re_{is the Reynolds number defined with} respect to the width (W) of the cavity. This is solved along with the streamfunction equation (SFE) given by

@2w

@x2þ

@2w

@y2¼

x

ð7Þ

Impulsive start is assumed for the initial condition on

w

and

x

, while no-slip condition at the walls is used as the boundary condi-tion. The SFE (Eq.(7)) is solved using unpreconditioned Bi-CGSTAB method[19], while the time integration of VTE (Eq.(6)) is carried out using fourth order four-stage Runge–Kutta (RK4) method. In

all computations, we have used uniform grid with 257 points in thex-direction and 257⁄ARpoints in they-direction, whereARis

the aspect ratio (height by width,H/W) of the rectangular cavity. A time step ofDt= 1103_{is used for the reported simulations.}

The first and second derivatives of

x

in the VTE are evaluated using combined compact difference (CCD) scheme reported in

[18]. This scheme is used for its good diffusion discretization prop-erty and high spatial resolution as compared to other schemes[18]. General stencils of CCD scheme for an internal node are given by

7

where,uis a function defined on a domain ofNequidistant points with grid spacinghand primes indicate derivatives with respect to independent variables. Both the equations are used for j= 3 to (N2), whereNis the number of grid points. With Dirichlet bound-ary conditions atj= 1 andj=N, we have 2Nunknown derivatives. Out of these unknowns, the derivatives at j= 1 are given as: u0

1¼ ð1:5u1þ2u20:5u3Þ=handu001¼ ðu12u2þu3Þ=h2. Atj=N,

similar expressions are used for the second derivative, while for the first derivative, we have signs of the coefficients are changed on the right hand side. Forj= 2 andj=N1, the derivatives are andN, resepctively. One simultaneously obtains the first and second derivatives by numerical solution of the stencils along with the closure schemes given above. Advantages of this scheme are dis-cussed in[15,18]. Effectiveness parameters are defined based on the spectral representation of the first and second derivatives and their properties are studied in the appendix.

3. Flow in rectangular lid driven cavity

Here Navier–Stokes equation has been solved in (

w

,

x

)-formu-lation using CCD scheme for spatial discretization andRK4time

integration scheme. In Fig. 1, representative vorticity contours are shown forRe=U1W/m= 6000 att= 1187 to explain the

geom-etry and selected sampling points’ location with respect to the co-ordinate system chosen. This definition sketch is for the RLDC with AR=H/W= 2, withU1in the positivex-direction. We have usedW

andU1as the length and velocity scales and time is

non-dimen-sionalized byW/U1. Contours drawn by solid lines indicate

posi-tive vorticity and the dashed lines indicate negaposi-tive vorticity in all figures shown. The flow field is analyzed, with three points cho-sen atD1,D2andD3, where time histories of

x

are recorded. Point

D1is in the top cell at (x= 0.50,y= 0.75), while pointsD2andD3are

in the second cell at (x= 0.25,y= 1.25) and (x= 0.004,y= 1.996), respectively. Indicated jet inFig. 1is responsible for transferring momentum and energy from the top to bottom cells.

Figs. 2–4show time histories of vorticity (left frames) and cor-responding Fourier transforms (right frames) forD1,D2andD3.

Re-sults shown are forRe= 5757, 6000, 6250, 6300 and 6700. The time series at these points forRe= 5757 show an initial transient fol-lowed by temporal instability and subsequent non-linear satura-tion. Extensive simulations and Hopf bifurcation analysis[20,21]

for this geometry reveal a critical Reynolds number of 5752. Hopf bifurcation indicated in the time series is described in the follow-ing sub-section. The time variation is clearly multi-periodic, as noted in the FFTs. ForRe= 5757, one notices absence of a single dominant mode forD1. However forD2andD3, one notices a

dom-inant mode atf= 0.1253, while there are other secondary modes of non-negligible amplitudes. The dynamics is distinctly different for Re= 6000, as noted from the corresponding frames ofFigs. 2–4, which show essentially single-period dynamics. Specifically for the representative points atD2andD3, time variation is essentially

monochromatic withf= 0.5938, whileD1has an additional

super-harmonic atf= 1.184. The time series forRe= 6000 shows signifi-cantly reduced amplitude for bothD2andD3, implying very low

levels of disturbance in the second cell, while atD1 disturbance

is of similar magnitude, as forRe= 5757. With increase ofReup to 6250 and 6300, one notices the strengthening of the spiral chain of vortices in the second cell. While this is indicated in the time series, the FFT indicates the presence and strengthening of other

0 0.2 0.4 0.6 0.8 1

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distinct modes, as the Reynolds number is increased. However, as the Reynolds number is increased further, the flow suffers a qual-itative change, as evident from the vorticity time series and FFTs for the flow atRe= 6700. This flow shows multi-periodicity in both the cells, with higher levels of fluctuations, due to presence of mul-tiple harmonics with side bands. Thus the flow in the bottom cell, shows a quieter state for Re= 6000 with a single frequency, as compared to the flow forRe= 5757. ForRe= 6700, the flow is char-acterized bychaoticmulti-periodic time variations with prominent side-bands as shown in Figs. 2–4 – a characteristic feature dis-cussed for low dimensional dynamical systems depicting chaos and soft turbulence in[22].

The quieter state for Re= 6000 is studied further in Fig. 5, where the Fourier amplitude of the most dominant mode [Ae(f)] is plotted as a function of Re for the points at D1 (top

left), D2 (top right) and D3 (bottom). For D3, one notices a

range of Re around 6000, for which the flow is quieter. In contrast, the amplitude of vorticity for D1 shows significantly

higher variations till Re= 6300, as compared to the other points. The amplitude of the most dominant mode for D1

shows a relative decay with increase in Reynolds number be-yond Re= 6300. This does not mean that the disturbance amplitude in the top cell as a function of Rehas reduced. This is a manifestation of energy being distributed equally among multiple modes in the top cell, as is readily evident from

Fig. 2showing the presence of various modes for this Reynolds number. This once again reveals soft turbulence, which leads to enhanced mixing in the top cell and higher transport of energy to the bottom cells. Note that apart from the second cell from the top, there is an incomplete cell just below it attached to the bottom wall in Fig. 1.

Fig. 3.Vorticity time history (left frames) and the corresponding Fourier transform (right frames) shown for pointD2(x= 0.25,y= 1.25).

Fig. 4.Vorticity time history (left frames) and the corresponding Fourier transform (right frames) shown for pointD3(x= 0.004,y= 1.996).

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3.1. Multiple Hopf bifurcations

Detailed Hopf bifurcation analysis of flow inside square LDC was reported in[20,21]. It was shown that the time series of vor-ticity depicted initial transient after which the linear growth was followed by non-linear saturation. The vorticity perturbation in lin-ear growth region is expressed in terms of various instability modes using Galerkin expansion. The time-dependent amplitudes of these instability modes during linear growth are given by,

AjðtÞ ¼ ðconst:Þesjt, as described in[20]. Saturation of disturbance

beyond the linear region can be expressed by Landau-Stuart-Eck-haus (LSE) equation in[1,20,21]and references contained therein

dAj

dt ¼sjAjþNjðAkÞ ð12Þ

where Nj(Ak) accounts for all non-linear interactions among

different modes (including self-interaction). In contrast, in the Stu-art-Landau equation[1,20], only one dominant mode is considered withsj=

r

r+i

x

1, representing its linear complex temporal growth

exponent. For the non-linear interaction, Landau and Stuart consid-ered only the self-interaction term given by,Nj¼ 2lAjAj

2_{, where}

l=lr+iliis the Landau coefficient. WritingA=jAjeih, one can rewrite

Eq.(12)for the amplitude and phase of the single dominant mode (A) as

djAj2

dt ¼2

r

rlrjAj

4

ð13Þ

dh

dt¼

x

1

li 2jAj

2

ð14Þ

An equilibrium amplitude can be obtained from Eq. (13) as,

jAej ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2rr=lr

p

. This analysis by Landau and Stuart forms the basis of classical Hopf bifurcation. For steady state equilibrium flows, there is no vorticity variation with time and Hopf bifurcation takes the flow from quiescent initial state to a periodic state, after the on-set of starting the lid impulsively. Such bifurcation diagrams show-ing variation of (jAej) vsRehave been presented in[1,20]for square

LDC. These diagrams also depict multiple bifurcations, due to the presence of several modes, in contrast to the single dominant mode

assumption in Landau’s and Stuart’s analysis. Such complex bifurca-tions and multi-periodic vorticity variabifurca-tions are also noted here for the RLDC. Presence of multiple Hopf bifurcation requires the use of Eq.(12), as discussed in details in[1].

While the flow in the top cell becomes supercritical for Re= 5752, displayed behaviour of the dominant mode inFig. 5(i) for the top cell is indicative of multiple bifurcations discussed above. However beyondRe= 6300, non-linearity and chaotic nat-ure of the flow takes over in the top cell. The second cell receives its energy from the top cell by the indicated impinging jet from right to left in the intervening space between these two cells. At the same time, this jet also shields the lower cell from the top cell events, as evident fromFig. 5(ii) forD2. This point also displays

pri-mary Hopf bifurcation at the same Reynolds number of 5752. IncreasingReabove this critical value, one notices a very coherent structure forming in the second cell, while the disturbance field in the second cell dramatically decreases, as seen inFig. 5(ii). When Re> 5800, another instability and bifurcation is noted in the sec-ond cell, as seen in the sub-figure. To begin with, this instability is characterized by a single-period phenomenon, as shown in

Fig. 3forRe= 6000 atD2. However this instability is global in

nat-ure, as can be seen fromFig. 2in the top cell. Non-linearity of dis-turbance field in the top cell creates the superharmonic seen for Re= 6000 inFig. 2. The dominant mode shown inFig. 5(ii), causes another bifurcation to start forRe’6250, resulting in breakdown of the coherent spiral chain of vortices in the second cell. Once again, such an event testifies the presence of multiple Hopf bifur-cations[1,20].Fig. 5(iii) indicates the dynamics in the third incom-plete cell attached to the bottom wall, which also reflects global dynamics of the flow inside the cavity. For example, formation of the coherent spiral chain of vortices quietens the disturbance field globally. This is seen inFig. 5(iii) by the constant amplitude dom-inant mode centered aroundRe= 6000. Instability of this chain of vortices is also recorded via the growth of the dominant mode in

Fig. 5(iii) beyondRe= 6300.

InFig. 6, flow topologies inside the RLDC are shown by vorticity contours forAR= 2, for the indicated Reynolds numbers att= 1187, after the initial transients have disappeared. ForRe= 5757, multi-cellular vortical structures are seen in the top two cells. In the top-Re

Ae

(f)

6000 6500

10-2 10-1 100

101 (i) D₁

Re Ae

(f)

6000 6500

10-4 10-3

10-2

10-1

(ii) D₂

Re Ae

(f)

6000 6500

10-5 10-4 10-3

10-2 _{(iii) D}

3

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most cell, gyrating orbital vortices with negative sign are noted without any central core. Motion of the lid induces a recirculating cell in the top, whose bottom edge creates a jet-like structure transporting conserved variables from right to left. The impinging jet on the left wall also creates the middle cell with vortices, which trickles down from the site of the jet impinging on the left wall. The motion of these vortices are very slow, as compared to the gyrating vortices in the topmost cell. ForRe= 6000, one notices dis-tinctly different and almost quasi-steady structures, with coherent vortices on a spiral up to the center of the middle cell. This is due to a unique equilibrium between convection and diffusion, as will be discussed in the following section. Vorticity contours forRe= 6700 inFig. 6(iii) are characterized by the absence of the coherent spiral vortical structure in the middle cell. The enhanced energy con-vected by the impinging jet, destabilizes the coherent spiral chain of vortices in the middle cell, indicating a departure from an inter-mediate equilibrium state.

3.2. Universal equilibrium state for RLDC

The unique feature of flow in RLDC attaining an intermediate equilibrium state withAR= 2 forRe= 6000, is also noted for RLDC with higher aspect ratios shown inFig. 7with vorticity contours, forAR= 2.5, 3 and 4 shown att= 1190. Despite minor differences in the top cell, the second cell shows identical coherent spiral vor-tical structures, as inFig. 6forAR= 2. Thus, the coherent spiral vor-tex chain in the second cell is a universal flow feature for RLDC withARP2 for a narrow range ofRearound 6000. However,

in-crease inARcauses the incomplete bottom cell to evolve into a full cell, while another incomplete cell attached to the bottom wall makes its appearance, as seen inFig. 7.

4. Enstrophy transport equation

The role of enstrophy in fluid flow to create rotationality is sim-ilar to kinetic energy describing the translational motion in fluid flow. While vorticity also describes rotationality, a measure of it is naturally obtained via enstrophy unambiguously describing the energy expended by the system in creating and sustaining rota-tionality. In all flows, physical instabilities take an equilibrium state to another and in the process, the energy of the system is redistributed into rotational and translation degrees of freedom. For example, in the present flow inside RLDC, the input transla-tional energy is partly converted into rotatransla-tional form and this will be manifested in creating enstrophy. Thus, enstrophy is a natural

dependent variable to study transitional and turbulent flows. We explain instabilities and pattern formations in RLDC, with the help of enstrophy transport equation (ETE) derived from the non-dimensional VTE in tensor notation given for 3D flows by

@

x

i

where subscripts,i,j= 1, 2 and 3, represent Cartesian axes and re-peated index implies summation. Taking a dot product of Eq.(15)

with

x

iand usingX1=

x

i

x

ito represent the local enstrophy, one

obtains its transport equation as

@

X

1

The third term on the left hand side (LHS) is due to vortex stretching (corresponding to the first term on the right hand side (RHS) of Eq.

(15)), which is absent for 2D flows. The diffusion of

x

igives rise to

RHS terms in Eq.(16). This contains a diffusion ofX1(the first term

of RHS) and the second term of RHS represents strictly a loss or dis-sipation term for the transport ofX1.

The present study views enstrophy as a point property in the flow domain and is different from the traditional approaches[6], where the enstrophy over the full domain is traced. The traditional approaches utilize the simplification brought about for problems which are homogeneous and hence periodic. However, such restriction to periodic flows leaves out most of the practical prob-lems which are inhomogeneous and the present study is for those class of problems.

Focusing on the case of 2D RLDC flow with the non-zero compo-nent of vorticity (

x

), the ETE can be written in vector form as

DX1

We note that the first term on RHS of Eq.(17)is missing from Eq.

(1), due to periodicity of the flow. Strictly negative RHS in Eq.(1)

implies the action of viscosity to dissipate energy globally. In contrast for inhomogeneous flows, the first term on RHS of Eq.

(17) can be either positive or negative. Therefore, the diffusion term in ETE can create or destroy rotationality, determined by the sign of RHS in Eq.(17). The two terms on RHS taken together determine net growth or decay ofX1. Thus, the diffusion term of

the VTE should not be identified strictly as dissipative for general flows. Diffusion and dissipation have been used interchangeably

[7], which is true for homogeneous flow in the global sense. In ETE, X1 is strictly positive and then RHS being positive would

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indicate the diffusion to cause local instability. It would act as a sink ofX1, where RHS is negative. This provides a mechanism of

creating rotationality at different scales by diffusion and is distinctly different from the concept of creating smaller scales by vortex stretching, as the only dominant mechanism of gener-ating small eddies relevant only for 3D flows. We also note that this role of diffusion in creating new length scales is ubiquitous for flows in both 2D and 3D.

To explain the mechanism of enstrophy creation at multiple scales simultaneously, one can plot the right hand side of the ETE. InFig. 8, the regions where the RHS of Eq.(17)is positive

are shown by dark shades for Re= 5757, 6000 and 6700 at t= 1187 for the RLDC withAR= 2. The blank regions in the figure indicate zones where RHS is negative, i.e., where diffusion leads to loss ofX1. Snapshot forRe= 6000 reveal the unique spiral

vorti-cal chain to form exactly at those places where RHS is positive, indicating the correspondence of regions where enstrophy is created by diffusion as explained by ETE. Creation of such coherent struc-tures, simultaneously quietens the flow in the neighborhood of the spiral, as indicated inFigs. 4 and 5for the representative point D3. This unambiguously establishes that at selective Reynolds

numbers, diffusion assists in creating rotational flow structures. x

y

0 0.5 1

0

0.5

1

1.5

2

2.5

(i) AR = 2.5

x

y

0 0.5 1

0

1

2

3

4

(iii) AR = 4

x

y

0 0.5 1

0

0.5

1

1.5

2

2.5

3

(ii) AR = 3

Fig. 7.Vorticity contours forRe= 6000 plotted for RLDC flow with (i)AR= 2.5, (ii)AR= 3 and (iii)AR= 4 att= 1190.

x

y

0 0.5 1

0

0.5

1

1.5

2

(i) Re = 5757

x

y

0 0.5 1

0

0.5

1

1.5

2

(ii) Re = 6000

x

y

0 0.5 1

0

0.5

1

1.5

2

(iii) Re = 6700

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To further investigate effects of diffusion in Eq.(17)at multiple scales, one can derive transport equations for higher powers ofX1.

Multiplying Eq.(17)withX1and definingXn¼X2

n1

1 , one obtains

forX2the following transport equation

DX2

Noting further thatDX2 Dt ¼2X1

DX1

Dt, one can write Eq.(18)as the ETE,

i.e.. an evolution equation forX1. Multiplying Eq.(18)withX2and

simplifying one can obtain transport equation forX3, which can be

used to write the ETE involvingX1,X2andX3. This process can be

generalized to obtain the transport equation forXnas

DXn

andX0=

x

withPindicating summation over allk’s andQ

indicat-ing the product of all thejth elements.

Also, the substantive derivative ofXncan be written and

simpli-fied as

which can be further simplified to give

DXn

as the ETE given by

DX1

One notes that while writing the transport equation forXn, the

dif-fusion term from the transport equation forXn1contributes two

terms; one of which is strictly dissipative (dependent on Xn2)

and the other as a diffusion term for Xn1. The diffusion term

involving Xn1 can be furthermore expressed into two terms

involving a strictly dissipative term withXn1and another diffusive

term involving Xn2. This process can cascade indefinitely in Eq.

(22), for increasingn with the leading term as a diffusion term and the rest are strictly dissipative. Higher order moments of ens-trophy will contribute more for higher wavenumbers, implying that

the order of even moments of enstrophy will be restricted by the energy supplied to the flow.

For 3D flows as well, the RHS of Eq.(22)is present as the forcing term. However, in this case, the vortex stretching term is retained. The ETE for 3D flow is same as given by Eq.(16). Following a sim-ilar approach as in deriving the transport equation forXnfor 2D

flows, the transport equation forXncan be derived for 3D flows

to be given by

where expression forCis same as in Eq.(20). Using Eq.(21)one can rewrite the ETE for 3D flows as

DX1

One notes that the diffusion term gives rise to the enstrophy cas-cade for both 2D and 3D flows, for which the contribution at higher wavenumbers depends upon the value ofn, decided by the energy supplied to the fluid dynamical system. However in 3D flows, vor-tex stretching is also present which provides an additional mecha-nism of energy redistribution process. This indicates that in 3D flows, generation of different scales of vorticity is due to enstrophy cascade via the diffusion term and energy cascade is via the vortex stretching which is implicit in convection process. In 2D flows, it is only the diffusion term which gives rise to enstrophy (and hence vorticity) at different scales.

5. Requirements on diffusion discretization

Accurate diffusion discretization is very important as it helps in controlling aliasing[18]. This is also important from the point of view of determining correct dynamics of the enstrophy. For the present case, the VTE given by Eq.(6)can be integrated over the whole cavity and using Gauss’ divergence theorem, one can show that the convection term has zero contribution, as shown below

Z Z

r

ð~V

x

Þdx dy¼ I

ð~V

x

Þ n dl^ ¼0 ð25Þ

where,^nis the unit normal vector at the boundary anddlis the ele-mentary tangential length along the boundary. Hence, the time rate of vorticity integrated over the whole domain is only dependent upon the diffusion of vorticity integrated over the whole domain.

7

200 400 600 800

(i) Contours of F

0.014

200 400 600 800

(ii) Contours of FU

B A

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The diffusion of vorticity (r2

_x

_{) integrated over the whole 2D}

do-main can be written, again using Gauss’ divergence theorem as

Z Z

If the vorticity is represented in spectral plane as,

x

ðx;yÞ ¼R R

Uðkx; kyÞeðikxxþikyyÞdkxdky, then the above equation

can be simplified for the RLDC withAR= 2 to obtain

Z Z

the quantityFUare plotted in frames (i) and (ii) ofFig. 9, respec-tively, in the (kx,ky)-plane. One can perform similar analysis for

anyARcase, the present analysis forAR= 2 is simply for the purpose of illustration, where we consider flow forRe= 6000 at t= 1480. One notes that the factorFin frame (i) is maximum when bothkx

and ky are maximum. This immediately suggests that one must

choose a method which does not filter out these high wavenumber components, as also evident from frame (ii) ofFig. 9. These high wavenumber combinations give rise to contributions to higher ali-asing error, the region marked above the dotted line AB. This high-lights that the numerical scheme must be able to resolve high wavenumber components in (kx,ky)-plane, as the method of[18].

Here, the sameCCDscheme has been used, which accurately cap-tures the high wavenumber components. How different numerical methods handle diffusion discretization is discussed in the appen-dix, by comparing theCCDscheme with second order central differ-ence scheme (CD2) as a reference. It is noted that many

conventional schemes for diffusion discretization do not have the ability to represent diffusion operator accurately.

6. Summary and conclusion

The role of diffusion and its relation to creating enstrophy (or rotationality) during flow instabilities and forming patterns are investigated here for an inhomogeneous flow. We have chosen the problem of flow inside RLDC with definite boundary conditions representing an inhomogeneous flow shown in Fig. 1. We have used results obtained by DNS of flow inside the RLDC to show that diffusion is responsible for forming a pattern of spiral vortices in the second cell from the top, at a universal Reynolds number of Re= 6000, as noted inFigs. 1 and 6for aspect ratio,AR= 2. The rea-son for formation of the special flow structures is explained with the help of vorticity time series inFigs. 2to 4, and its Fourier trans-form. It is also noted that following the first Hopf bifurcation for the RLDC with AR= 2 for Re= 5752, the flow exhibits a narrow range of Reynolds number around 6000, where the second cell of the cavity is dominated by the coherent pattern formation with low levels of disturbance elsewhere, as noted inFig. 5. Role of sub-sequent bifurcations in creating chaotic flow with multi-periodic disturbances are also noted forRe= 6700, which is indicative of soft turbulence in internal flows[20,22]. The bifurcation analysis and the study of instabilities in this work are carried out based on the solution of full Navier–Stokes equation and not on linear-ized theories.

The spiral vortex chain is also shown to exist for higher aspect ratio of the cavity, shown for AR= 2.5, 3 and 4 in Fig. 7 for Re= 6000. This implies a universal equilibrium state determined by the supply of energy (represented byRe) and not on the geom-etry (represented byAR) of the cavity. To explain the creation of rotationality by the diffusion operator, we have developed a trans-port equation for enstrophy (ETE) from the Navier–Stokes equation

to explain the true role of diffusion for general flows. Utility of the developed ETE is established by plotting the right hand side of this equation inFig. 8, which shows the direct correlation of the pat-terns formed by the rotating cells with the vorticity contours in

Fig. 6. For the flow in RLDC, unique equilibrium state is shown as a consequence of non-negligible diffusion along with convection in an intermediate post-critical range of Reynolds number around 6000. Organized enstrophy is shown as a consequence of over-rid-ing action of diffusion in creatover-rid-ing rotationality in this flow. Such equilibrium can be observed in numerical simulations, only when special care is exercised for diffusion discretization. In this work, we have also shown that diffusion and dissipation are not identical for inhomogeneous flows, as opposed to equating these for simula-tion results of homogeneous turbulent flows[7]. It is furthermore explained from ETE how different scales are created, shown by the various powers of enstrophy in Eqs.(20) and (22)for 2D flows and in Eq.(24)for 3D flows.

Thus, the present research was conducted with the sole aim of showing the physical and numerical aspects of diffusion process for the accurate simulation of Navier–Stokes equation for inhomo-geneous flows. It is emphasized that physically the role of diffusion for inhomogeneous flows is not strictly dissipative, as is the case for homogeneous turbulent flows. This is achieved by developing ETE, with enstrophy characterizing the rotational energy of a flow. By developing the ETE, in terms of higher even moments ofX1, we

identified the indexnin this equation, which is indirectly fixed from the total energy imparted to create the flow. This approach of viewing how smallest scale is fixed is entirely different to the lo-gic employed for the dissipation of kinetic energy to heat, as in explaining Kolmogorov’s scale.

In explaining the numerical significance of accurate discretiza-tion of diffusion operator, we have developed a balance equadiscretiza-tion for 2D flows given as

Z Z _@

_x

The relevance of diffusion operator compared to convection process is demonstrated by this equation, where we show that convection does not contribute at all due to the wall boundary conditions, while the instantaneous rate of change of vorticity in the full do-main (LHS of Eq.(28)) is completely determined by the diffusion process (given by the RHS of Eq.(28)). The importance of diffusion discretization carried over the whole domain is explained with the help ofFig. 9, which clearly establishes the role of high wavenum-ber components. This figure shows the inadequacy of conventional diffusion discretization term as compared to theCCDscheme em-ployed here.

Appendix A. Spectral analysis of numerical methods for DNS

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@u @tþc

@u

@x¼0; c>0 ðA:1Þ

Eq.(A.1)admits a unique non-dispersive and non-dissipative solu-tion, for which the group velocityVgis equal to the phase speed (c).

Considering a grid with uniform spacing ofh, the unknown quantity uat anyjth node can be represented by its Fourier transform given by

uj¼uðxj;tÞ ¼ Z

Uðk;tÞeikxj_dk _ð_A_:₂_Þ

The exact spatial derivative ofucan be obtained from above as @u

@x

exact¼

R

ikU eikxj_dk. When Eq.(A.1)is solved numerically by dis-crete methods, the spatial derivative (denoted byu0

j) can be

repre-In a finite-domain, the spatial derivativeu0_{can also be expressed as}

[23], fu0_{g ¼}1 and N is the total number of discrete nodes. Here, [C] matrix is dependent on the spatial discretization scheme for the first deriva-tive and this incorporates boundary closure schemes as well. As gi-ven in [23], essential numerical properties for space–time discretization schemes are: (i) Numerical amplification factor defined as, G¼Uðk;tnþ1_Þ

Uðk;tn_Þ; (ii) Normalized numerical group velocity, VgN/cand (iii) Normalized numerical phase speed,cN/c.

Using Runge–Kutta time integration schemes in conjunction with higher order compact schemes for spatial discretization pro-vides excellent numerical properties, as analyzed and shown in

[17,18]and used in[14,15]. Four-stage fourth order Runge–Kutta (RK4) method can be used to integrate equations of the form dy

ForRK4time integration, the numerical amplification factor at any

jth node is given by[23]

For the initial solution of Eq.(A.1)given by

uðxj;t¼0Þ ¼u0j ¼ Z

A0ðkÞeikxjdk ðA:6Þ

the general solution at any arbitrary timetn_{can be obtained as}

un

j ¼

Z

A0ðkÞ ½jGjjneiðkxjnbjÞdk ðA:7Þ

One concludes that the physical solution is neutrally stable, and

hence the amplitude ofGjis defined as, jGjj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

G2_rjþG2_ij

q

withGrj

and Gijdenoting real and imaginary parts. Ideally, jGjj should be

identically equal to 1, at all times for neutral stability. If one defines tan bj

¼ Gij

Grj, thennbj=kcNtdetermines the phase of the solution with cN denoting numerical phase speed. One notes that cN is

dependent onk, i.e.. the numerical solution is dispersive as opposed to the non-dispersive exact solution. From these relations, one ob-tainsVgN¼@x@kN[23]. Thus, one obtains expressions for the

normal-ized numerical phase speed and numerical group velocity as[23]

cN

Numerical properties are plotted in (Nc,kh)-plane in order to

com-pare different schemes based on their ability to accurately solve Eq.

(A.1). InFig. A.1, the contours ofjGj,cN/candVgN/care plotted for (a)

RK2-FS scheme: which uses two-stage, second order Runge–Kutta

(RK2) method for time integration and Fourier spectral method for

spatial derivatives and (b)RK4-CCD scheme: which usesRK4method

(a) RK

₂

-FS

(b) RK

₄

-CCD

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for time integration and spatial discretization by CCD scheme as de-scribed in Section2 [17,18].

One expects maximum resolution while using Fourier spectral method for spatial derivatives. However for space–time dependent governing equations, the performance of combined discretization schemes are more relevant, as highlighted in[23]. ThejGjcontours forRK2-FS graphically show this effect as the combined scheme is

unstable as shown in top left frame ofFig. A.1, i.e.jGj> 1 in the en-tire (Nc,kh) plane implying the amplitude of the solution of Eq.

(A.1) to increase unbounded with time for Eq. (A.1). However, when one looks at similar contours forRK4-CCD scheme, one notes

a small range ofNc(up toNc= 0.0576) for which the scheme is

neu-trally stable for all resolved values ofkh. The contours also reveal thatRK4-CCD scheme has jGj< 1 for 0.0576 <Nc< 1.3197. In the

middle frames, the contours forcN/chave been plotted for both

the schemes. A scheme capable of solving Eq. (A.1) accurately should not exhibit any error due to numerical phase speed, which will be the case in regions where cN/c1. One notes that for

RK4-CCD scheme, a region for small values of kh exists where

cN/c1. No such region is present forRK2-FS scheme. Also VgN/c

should be unity for all values ofkh, while solving Eq. (A.1). For RK4-CCD scheme, one notes that there exists a region with small

values of kh and Nc, where VgN/c= 1. No such region exists for

RK2-FS scheme.

The reason that the unstable scheme worked in[7]is because: (i) the numerical instability is very mild, which does not lead to catastrophic breakdown and (ii) the adopted method for solving homogeneous turbulence problem also uses hyperviscosity which stabilizes higher wavenumber components numerically. We also note that due to the numerically unstable nature of theRK2-FS

method, no explicit excitation is needed to trigger turbulence. The above analysis of the two schemes considered highlights the importance of choosing a scheme for DNS. One can easily conclude that RK4-CCD scheme is superior to RK2-FS scheme for solving

flows where convection is the dominant physical phenomenon. TheRK4-CCD scheme is not only useful for solving flows with

convection as the dominant phenomenon, but also for flows which require good diffusion discretization as is the case with present work. For studying the effectiveness of the CCD scheme in discret-izing first and second derivatives, one is required to carry out the spectral analysis as in[17,18]. Numerical evaluation of first deriv-ative gives rise to an equivalent wavenumberkeqas obtained in Eq.

(A.3). For investigating spatial resolution of a numerical scheme, we define a parameterkeq

k whose real part represents the numerical

method’s ability to resolve different scales. Ideally, this parameter should be equal to 1 for allkhup to the Nyquist limit which is the maximum limit on resolved wavenumbers. Similar effectiveness

parameter can also be defined for the accuracy of second derivative discretization.

In spectral representation, exact second derivative of the quantity

ucan be obtained as, @2u second derivative is obtained numerically, it is given by

u00

The real part of the parameterk

ð2Þ eq

k2 represents the ability of the

numerical scheme (CCD in this case) to effectively discretize the second derivative. The expressions forkeqandkð2Þeq are obtained from

the stencil of CCD scheme given by Eqs.8 and 9as also reported in

[17,18].Fig. A.2(a) shows the plot of real part ofkeq/kvskhwhile

Fig. A.2(b) shows the real part ofkð2Þ_eq=k2vskh. Here, the expres-sions are evaluated for a grid with 50 points, i.e..N= 50. The param-eters are plotted for the near boundary pointsj= 2 and 3 and for an interior node (j= 25) for theCCDscheme. The parameters are also plotted for the second order central difference (CD2) scheme, as

indicated in the figure. One can note an exact scale resolution till aroundkh= 1.6 forj= 25 inFig. A.2(a) by theCCDscheme. For the CCDscheme, the second derivative is also exactly represented till aroundkh= 1.6 forj= 25 inFig. A.2(b). One notes fromFig. A.2that resolution of the CD2 scheme for both the first and second

derivatives is quite inferior to theCCDscheme. For second deriva-tive evaluated usingCCDscheme, there is an overshoot present at higher wavenumbers. This overshoot helps in eliminating numeri-cal instabilities arising due to aliasing error or any other numerinumeri-cal non-linear phenomenon which occurs at higher wavenumbers. This provides a highly accurate diffusion discretization, which is very important as has been shown in the present work that diffusion leads to creation/destruction of rotationality. This allows capturing vortices of almost all scales accurately and usage ofRK4scheme for

time integration allows to obtain a highly accurate solution whose accuracy is also established for a square lid driven cavity flow reported in[18].

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