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Vortical structures and wakes of a sphere in homogeneous and density stratified fluid * Liu-shuai Cao1, Feng-lai Huang1, Cheng Liu1, De-cheng Wan1, 2 1. Computational Marine Hydrodynamics Lab (CMHL), School of Naval Architecture, Ocean and Civil Engi- neering, Shanghai Jiao Tong University, Shanghai 200240, China 2. Ocean College, Zhejiang University, Zhoushan 316021, China

(Received April 9, 2021, Revised April 11, 2021, Accepted April 12, 2021, Published online April 28, 2021) ©China Ship Scientific Research Center 2021 Abstract: Vortical structures and wakes of bluff bodies in homogenous and stratified environment are common and important in ocean engineering. Based on the Boussinesq approximation, a thermocline model is proposed to deal with the variable density stratified fluid, and implemented in the commercial software Simcenter STAR-CCM+ framework. The improved delayed detached eddy simulation (IDDES) modeling method is adopted to resolve the coherent vortical structures and turbulent wakes precisely and efficiently. Four conditions consisting of one homogenous and three stratified fluid cases with different density gradient past a sphere at Reynolds number 3 700 are investigated. Results show that density stratification has a great impact on the vortical structures, the vertical motion is suppressed and internal waves will be induced and propagated, which is very different with that of homogenous situation. With the stratification strength increases, the vortical structures are gradually flattened, the asymmetry and anisotropy between vertical and horizontal motions are enhanced. Key words: Vortical structures, wakes, sphere, homogenous, stratified fluid

Introduction The vortical structures and wakes behind a bluff

body have been a conical problem of interest for over a century. As a representative geometry of bluff body, the wakes behind a sphere in an unstratified or homo- genous fluid have been studied extensively by approaches of physical experiments[1-4] and numerical simulations[5-10]. The wake can be broadly broken down into two flow regimes depending on whether the boundary layer remains laminar up to separation (subcritical regime) or becomes turbulent before separation (supercritical regime). The Reynolds num- ber defined as = /Re UD , where U and D are characteristic velocity and length scales and is the dynamic viscosity, is often used to distinguish between two regimes. The Reynolds number describes the ratio of inertial forces to viscous forces, helps to

* Projects supported by the National Key Research and Development Program of China (Grant Nos. 2019YFB1704200, 2019YFC0312400), the National Natural Science Foundation of China (Grant Nos. 52001210, 51879159). Biography: Liu-shuai Cao (1990-), Male, Ph. D., Assistant Professor, E-mail: liushuaicao@sjtu.edu.cn Corresponding author: De-cheng Wan, E-mail: dcwan@sjtu.edu.cn

predict flow patterns in different fluid flow situations. At low Reynolds numbers, flows tend to be dominated by laminar flow, while at high Reynolds numbers flows tend to be turbulent.

Most of the studies using numerical simulation have been conducted for flow over a sphere in the subcritical regime by using several different turbu- lence modeling methods, such as direct numerical simulation (DNS)[6], large eddy simulation (LES)[5], partially averaged Navier-Stokes (PANS)[7], detached eddy simulation (DES)[9-10], and even Reynolds ave- raged Navier-Stokes (RANS) simulations[11]. These studies highlight several key features of flow past a sphere in this regime: a stagnation point forms at the front of the sphere, a thin laminar boundary layer is formed on the surface of the sphere which grows until smooth separation occurs, an axisymmetric free shear layer, a roll-up of detached shear layers due to the Kelvin-Helmholtz (KH) instability, a vortex shedding in a three-dimensional turbulent wake, as well as a recirculating region directly behind the sphere.

While in the real environment on earth, the atmosphere and oceans are typically stably stratified, so the performance and efficiency of ships and sub- marines are inevitably influenced by density gradient. However, in most of the previous study, density stratification is ignored by engineers for the simpli-

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http://www.jhydrodynamics.com Journal of Hydrodynamics, 2021, 33(2): 207-215

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208

city. Stratification is known to have a large impact on

the wake behind a bluff body. Compared with the homogenous case, density stratification introduces several new concepts and processes, such as internal wave propagation and reflection[12-13], drag enhance- ment[14-15], and pancake eddies in the late wake[16-17]. The radial symmetry of the problem is broken and stratification disrupts motion in the vertical direction. A complex coupling between kinetic and potential energy occurs, and the system can support internal gravity waves which transfer momentum from the wake to the background. For a turbulent wake, stratification increases the lifetime of the wake and preserves both the mean velocity and vortical struc- tures into the late wake[18].

For wakes in stratified fluid, the Reynolds number alone is not enough to describe the flow behavior. The strength of stratification is characterized by an internal Froude number, hereafter referred to simply as Froude number, defined as = /Fr U ND ,

where 0= ( / )( / )N g z is the buoyancy

frequency with g the acceleration due to gravity,

0 the characteristic density, and / z the background density gradient[19]. According to the definition, the smaller Fr means the stronger stratification. In the presence of stratification, both the Reynolds number and Froude number must be specified to characterize the wake dynamics.

At the beginning, plenty of experiments were used to study the vortical structures and wakes of a sphere in stratified water tank[20-23]. With the development of computational methods and power, numerical methods have been proposed to resolve the stratified turbulence[24-27]. These studies revealed the characteristics of the stratified wakes. When Fr 1.00 , the vertical motion is almost completely inhibited and the flow is constrained to flow horizontally around the body. At 1.00Fr , the flow can be thought of as behaving like an unstratified wake. At intermediate Froude numbers there are significant differences between cases with the wake evolution dependent on both Re and Fr . The wake can be thought of as changing from a quasi-two- dimensional wake bounded by lee waves for

< 0.50Fr to a saturated lee wave regime for 0.40 << 0.75Fr to a transition regime where the wake

transitions from being dominated by the lee wave to one virtually unaffected by stratification for 0.75 <

< 2.25Fr to a near wake that behaves as an unstratified fluid at > 2.25Fr .

However, most of the numerical simulations are based on in-house codes, and the turbulence modeling methods are mainly DNS and LES, of which the computational cost are very expensive. Thus, in this

paper, based on Boussinesq approximation, a thermo- cline model is proposed and implemented in the commercial software Simcenter STAR-CCM+ frame- work. The main objective of the paper is to pave the way to the complex geometry and high Reynolds number simulation in industrial application, thus both homogenous and stratified fluid can be solved.

The paper is outlined as follows. Firstly, the thermocline model is proposed to predict flow field in linearly stratified fluid. The numerical simulation methods including computational domain, discretized mesh, turbulence modeling, initial and boundary conditions and solvers setup are given. Then four simulations consisting of both non-stratified and stratified cases are performed. The vortical structures, internal waves as well as density variations are analyzed sequentially. Lastly, the conclusions are drawn. 1. Numerical methods 1.1 Governing equations

The objective of this research is to simulate the linearly stratified fluid past a sphere. Unlike the homogenous fluid case, the density of water now

is also a variable. Besides the continuity and momen- tum equations under Boussinesq approximation, the heat equation is simultaneously solved to close the Navier-Stokes equations. The governing equations are shown as below:

= 0 u

(1)

21+ ( ) = + +P T

t

u

u u u g f+

(2)

2+ ( ) + =z

TT u T

t

u

(3)

where u , , P , , , g , T , f , denote the

velocity, density, modified pressure, kinematic viscosity, thermal diffusivity, volumetric expansion coefficient, gravitational acceleration, temperature fluctuation, and large-scale force required to maintain the turbulence, respectively. Moreover, zu and

(= d / d )bT z are the vertical component of fluid

velocity and constant background temperature gra- dient, respectively. 1.2 Thermocline model Pycnocline refers in general to density, but the corresponding nomenclature can be alternatively used to indicate a change due to salinity (halocline) or temperature (thermocline). In the open sea, the

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variation of salinity can be ignored when referring to the fluctuation of temperature. Therefore, in this paper, we mainly focus on the thermocline, which means that density can be simply specified as a function of temperature. For the top 40 m-500 m of the sea, we can describe density as a function of temperature

T in the form of a polynomial expression. While the temperature ranges as a function of the depth z below the free surface. Thus, in this paper, the density and temperature T can be written as:

0 1= ( ) = +T f z T c z , 0 2= ( ) = ( ) =T g z c z

(4)

where 0T , 0 are the reference temperature and

density, 1c and 2c are the temperature and density

gradient, respectively. 1.3 Computational domain and mesh

The case of flow past a sphere for fluid systems which are continuously (linearly) stratified is the focus of the present research. The physical system con- sidered is sketched schematically in Fig. 1. Fig. 1 (Color online) Computational domain

A liquid of mean density 0 and constant

buoyancy frequency N is in uniform rectilinear motion U past a sphere of diameter = 0.025 mD .

The density increases from top to bottom and a function of depth z . The flow takes place in a tank of length 12.5D , with 2.5D upstream, and 10.0D downstream, width 5.0D for a half domain, and depth 5.0D with 2.5D upward and downward, respectively. Incoming velocity U is determined by the Reynolds number specified both in non-stratified and stratified conditions, respectively. The block-based structured mesh is built using the commercial software ICEM CFD. A very carefully designed O-grid topology is applied around the sphere. The inner zone near the body, extending to about 1.6D outward distance is treated as a boundary layer refinement area, designed to capture the boundary layer and flow separation, the outer zone is used to provide a fine mesh density to resolve the flow field

and offer a balance of resolution and processing time. The total number of cells is 15 096 000. Details of the mesh is depicted in Fig. 2. Fig. 2 (Color online) Details of the mesh 1.4 Turbulence modeling DNS is too expensive in most engineering applications to resolve all the scales in time and space. As for RANS equations, they are suitable for many engineering applications, and typically offer the level of accuracy required and provide the most economical approach, but lack of resolution for the vortical structures with different scales. So, with the aim of reducing the impacts of the averaging procedures on turbulence and resolving at least a portion of the turbulence, the hybrid RANS/LES approaches based on the original shear stress transport (SST) -k model is utilized in this paper. The turbulence kinetic energy k and specific dissipation rate transport equations of SST -k turbulence model are as follows:

( )( )+ = + + +i t

k k kj j k j

kuk kP D S

t x x x

(5)

( )( )+ = + + +i t

j j j

uP D

t x x x

+CD S

(6)

where k and are the turbulent Prandtl num-

bers for k and . kP and P are the productions

of k and , respectively. kD and D are the

dissipations of k and , respectively. CD

denotes the Cross- Diffusion term. Sk and S are the

user-defined terms. The basic hybrid RANS/LES model is the DES model, which achieves the switch between RANS and LES by a comparison of the turbulent length scale tL with the grid spacing max .

The DES has many extensions due to the hybrid with different RANS turbulence models. The DES with SST -k model is derived by modifying kD as

DES=kD k F

(7) where DESF is expressed as

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DESDES max

= max ,1tLF

C

(8)

In the expression, DESC has a value of 0.61, max is

the maximum local grid spacing, is a constant

value of the SST model and tL is the turbulent

length scale

=t

kL

(9)

An updated version of DES, named improved DDES (IDDES), is provided by Shur et al.[28], which extends the LES zone of the model to the outer part of wall boundary layers. Similarly, for the IDDES, a parameter IDDESF is defined as follow:

RANSIDDES

IDDES

=l

Fl

(10)

where

RANS =k

lc

(11)

IDDES RANS LES= (1+ ) + (1 )d e dl f f l f l

(12)

where the LES length scale is redefined as:

LES DES=l C

(13)

max max= min[0.15max( , ), ]d

(14)

where df is the empirical function, ef is the

elevating function. 1.5 Boundary and initial conditions

There are four boundary conditions utilized in the simulation. As depicted in Fig. 1, the upstream and the top and bottom boundaries are treated as velocity inlet boundaries on which velocity components are imposed. The downstream boundary is specified to be outflow and the two side boundaries are symmetric planes. No-slip condition is specified for the sphere body. As for the entire domain with linearly thermal and density stratification, user-specified field func- tions are introduced to set the temperature and density, which are linear functions with z as defined by Eq. (4). Consequently, the initial pressure field is also user-defined as a function that varies with density and depth.

After initialization of the simulation, the middle symmetry plane ( = 0)y of temperature and density

are revealed in Figs. 3, 4, respectively. The two contours reveal that the distribution of temperature and density match exactly as the linear functions predefined, which means that the boundary and initial conditions are correctly imposed. Fig. 3 (Color online) Temperature distribution of = 0y plane

after initiation Fig. 4 (Color online) Density distribution of = 0y plane after

initiation 1.6 Solvers and computational matrix

The simulations in this paper are performed with the commercial software Simcenter STAR-CCM+. It is a powerful solution for solving multidisciplinary problems in both fluid and solid continuum mechanics within a single integrated user interface. The CFD solver uses a cell-centred finite volume discretization applied to cells of arbitrary polyhedral shapes, and offers a selection of turbulence models suitable for a wide range of practical applications, for example, ship hydrodynamics, aerodynamics and innovative equip- ment prototype design. The computational matrix for the simulation is listed in Table 1. Four simulations that = 3 700Re ,

while =Fr 3.00, 2.00, 1.00 represents different stratification levels, and one unstratified simulation,

=Fr , are performed. Table 1 Computational matrix

Re Fr 3 700 3 700 3.00 3 700 2.00 3 700 1.00

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2. Results and discussion

2.1 Coherent vortical structures The three-dimensional visualization of vortical

structures can be expressed by -Q criterion which de-

fines a vortex by the region where the rate of rotation tensor ij exceeds strain rate tensor ijS [29-30].

2 21= ( )

2Q S ,

1=

2ji

ijj i

uu

x x

,

1= +

2ji

ijj i

uuS

x x

(15)

Figure 5 visualizes instantaneous vortical struc- tures in the wake using the -Q criterion at = 1Q for

the homogeneous case. It shows that vortex rings are shedding from the sphere, and remain circular before breaking down at around 2.5x D . Further down- stream, transition gradually begins, these vortical structures turn to be branches of entangled tube-like structures with a high length-to-diameter aspect ratio, which is called the vortex tube. These vortex tubes have no directional preference, and the density of vortex tubes per unit volume decrease significantly after 6.0x D . Fig. 5 Iso-surface of = 1Q for case =Fr . Note that the

sphere center is at = 0x , = 0y and = 0z

Figure 6 shows vortices at magnitude of = 1Q

in four different cases, =Fr , 3.00, 2.00 and 1.00, respectively. It is obvious that with the Fr decreases, which means that the stratification strength levels enhanced, vortex tubes are tended to be confined to the streamwise-oriented region. The vortical structures in = 1.00Fr turn to be flattened and break into smaller coherent vortical structures downstream. It is interesting to find that the density of vortical struc- tures in the stronger stratification ( = 1.00Fr ,

= 2.00Fr ) are higher than the unstratified case ( = )Fr . This indicates that vortices shed from the

body are not sole sources of the vorticity, maybe internal waves induced by the stratified fluid is another reason responsible for the high-density vortex structures in the wakes. Fig. 6 Iso-surface of = 1Q for four cases =Fr , 3.00, 2.00

and 1.00

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Higher magnitude of = 100Q in four different

cases are shown in Fig. 7. Since Q represents region

where the rate of rotation tensor ij exceeds the

strain rate tensor ijS , a high value of Q signifies

intense rotation of fluid elements. At this level of Q ,

the downstream vortical structures are present mainly in the region 1.5 < / < 5.0x D . The vortical structures depicted by the higher Q for weak stratification

( = 3.00)Fr bear some similarities to the

homogeneous case ( = )Fr in term of the density

of vortex tubes, aspect ratio and transition orientation. However, the downstream vortical structures are significantly different in the stronger stratification ( = 1.00)Fr case. There is a pile of long thin

flattened tubes that alternate on either side of the vertical center-plane. Compared to =Fr case, the cross section of the vortex tube at = 1.00Fr changes from circular shape to elliptical shape, as shown in the insets.

After we investigate the coherent vortical struc- tures in different Fr cases, we can conclude that the vortex rings are shed from the sphere, and remain circular near the body in both stratified and unstratified cases. In the homogenous case, these vortical structures become a bundle of vortex tube structures with circular cross section. While in the strongly stratified case = 1.00Fr , there is a strong suppression in the vertical direction so that the original tube-like form structures turn to be flattened. The asymmetry and anisotropy are enhanced between the vertical and horizontal directions.

2.2 Internal waves As stated before, the shape and density of the

vortical structures downstream in the stratified fluid is related to internal waves. According to Spedding[18], the projection of the motions on horizontal plane { , }x y gives estimates of the two horizontal velocity

components, = { , }u vq , and their gradient quantities

z and z . These two parameters are derived to

reveal the nature of the flow. The magnitudes of the vertical vorticity z and the divergence field z

can be regarded as approximately proportional to the amplitudes of vortex and wave motions in a plane of constant z , especially z can be used as an indi-

cator of internal waves.

=z q , =z q

(16)

A horizontal plane of = 0z is chosen and rendered by the vertical vorticity z in Fig. 8 and

Fig. 7 (Color online) Iso-surface of = 100Q for four cases

=Fr , 3.00, 2.00 and 1.00. Inset on lower right corner shows the cross-section variation of the vortex tube

213

Fig. 8 (Color online) The contour of the vertical vorticity z

in the plane of = 0z for three stratified cases

Fig. 9 (Color online) The contour of the internal waves z in

the plane of = 0z for three stratified cases

divergence field z in Fig. 9, respectively. As for

the vertical vorticity contours, the range becomes broader with the decrease of Fr . It relates to the vertical motion suppression discussed in Section 2.1. The vortices are inhibited in vertical direction, and squeezed to the horizontal direction, which makes it higher in horizontal plane. After some time evolving, internal waves are emitted and propagated in the wake. These internal waves spread out and have wider spread zone as the Fr decreases. Unlike the homogeneous case, these internal waves will change

the vortical structures downstream which is discussed above. 2.3 Vertical motion suppression and density fluctua-

tions in the thermocline In this paper, there is a density gradient exists in

the vertical direction expect for the homogeneous case. Fig. 10 shows the density isopycnals for three di- fferent cases. The pictures show that the presence of the sphere distorts the original stable distributed horizontal isopycnals, mixing the fluids up and down over a distance much longer than the sphere diameter. The vertical influencing zone in the z direction behaves differently. Since the higher Fr relates to the lower density gradient, the vertical influencing height reaches the maximum value to 2.6D , while in the strongest stratification case = 1.00Fr , the vertical influencing height is only 1.8D . As a result of vertical motion suppression, in the strongly stratified fluid, the wake tends to grow horizontally and break into horizontal vortices which is consistent with the previous discussion.

Fig. 10 (Color online) Density isopycnals in the middle symme-

try plane ( = 0)y for three different cases

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What’s more, the density fluctuations of different Fr cases at two specific locations / =x D 1.0, 5.0 are analyzed. From the Fig. 11, consistent conclusions will be easily achieved that as the Fr decreases, the degree of distortion of density layer decreases. This is because the higher density gradient, the stronger for the vertical motion suppression. Furthermore, to reveal the density fluctuations before and after the sphere, = 2.00Fr case is chosen as an example, and line 1 to line 7 are chosen as Fig. 10 shows. These lines are located at / =x D 1.0, 1.0, 2.0, 3.0, 4.0, 5.0 and 8.0 respectively. The density fluctuations on these lines are shown in Fig. 12. The line 1, which is located before the sphere can be a representative of the initial density distribution with the depth, so that the degree of distortion downstream can be evaluated and compared. It is of interest to find that line 2 to line 7 is distorted sharply around 1.0D to 1.0D in the z direction because of the presence of the sphere. However, at around 8.0D downstream, the density fluctuations of line 7 are smaller compared with the initial density gradient (line 1). Moreover, the heavier water will gradually float up and mixed downstream, which will contribute to the anisotropy of the vortex structures shown in Fig. 6.

Fig. 11 (Color online) Density fluctuations comparison at

/ =x D 1.0, 5.0 locations among different Fr cases

3. Conclusion The present work is devoted to a detailed inves-

tigation on vortical structures and wakes of a sphere both in homogenous and stratified fluid. A thermocline model method is proposed to deal with the stratified fluid simulation and implemented in the commercial software Simcenter STAR-CCM+. The vortical structures and wakes are well resolved near and downstream the sphere. Body-generated turbulent wakes and vortices, internal waves as well as density fluctuations, and vertical motion suppression caused by density stratification are analyzed. Results demon- strate that the stratification has a great effect on the flow field, which are very different from those of homogenous situations. This paper has established a first step towards the goal of predicting the vortices and wakes for a fully appended submarine operating in the real ocean environment. Further efforts can be devoted to improving the resolution of the predictions, where LES or DNS may be used.

Fig. 12 (Color online) Density fluctuations of different loca-

tions for = 2.00Fr case

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