Numerical study of the structure and dynamics of a

Journal of Wind Engineering & Industrial Aerodynamics 177 (2018) 306–326

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Numerical study of the structure and dynamics of a tornado at the sub-critical vortex breakdown stage Zhenqing Liu a, *, Heping Liu b, Shuyang Cao c a b c

School of Civil Engineering and Mechanics, Huazhong University of Science & Technology, Wuhan, Hubei, China Department of Civil and Environmental Engineering, Washington State University, Pullman, WA, USA Department of Civil Engineering, Tongji University, Shanghai, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Tornado LES Numerical simulation Breakdown Reynolds averaged force balance TKE budget Spectrum

A tornado at the sub-critical vortex breakdown was studied in detail using large eddy simulations. The averaged flow fields and the fluctuations were examined. Correlations of the fluctuating parameters were also studied, and a special turbulence structure was found. To clarify this turbulence structure, the Reynolds averaged force balances in two directions were studied, in which the sub-terms of the turbulence forces were paid more attention. The turbulence kinetic energy budget was examined to understand the balance between turbulence production, transportation, dissipation, and advection, among others. The time histories of wind velocities at some representative points were recorded, and three types of motions were then observed. The strength of these motions was furthered examined through spectrum analysis to study the change in them at different heights. Finally, the flow field visualization was performed to reveal the mechanism underlying these motions and explain some interesting phenomena found in the present study.

1. Introduction In recent years, many studies have been conducted to examine flow structures in tornados, most of which focus on time-averaged flow fields and the fluctuations. Some important findings have been obtained, such as the strong jet flow close to the ground, the low pressure in the core, and the Ekman-like spiral, etc. The most important parameter determining the flow structure was found and named as swirl ratio. By increasing this parameter, the tornado will experience several stages, i.e., single-celled vortex, vortex breakdown, vortex touchdown, two-celled vortex, and multi-celled vortex, as shown in Fig. 1. For the single celled vortex, the core extends upward from the surface to higher levels and spreads slightly. At the vortex breakdown stage, the flow changes from a tight, laminar vortex to a broader, turbulent state. The laminar flow with a very narrow core in the lower portion moves upward until it suddenly expands into a recirculation bubble. For the two-celled vortex, the inner core flow moves downward but the flow surrounding the core of tornado moves upward. And at the multi-celled vortex stage, a family of several secondary vortices rotating around the main vortex is evident. The different types of tornado like vortices have been reproduced successfully by various types of tornado simulators. The first tornado simulator was developed by Ward (1972) and then applied by many

researchers, i.e., Church (1977, 1979), Monji (1985), Diamond and Wilkins (1984), Mishra et al. (2008), and Matsui and Tamura (2009), etc. For this type of tornado simulator, the swirling of the fluid is provided by the rotating screen or guide vanes mounted on the ground. In the last decade, a new tornado simulator was developed at Iowa State University are then applied by Tari et al. (2010), Kikitsu et al. (2012), Cao et al. (2015), and Wang et al. (2016), etc. For this type of tornado simulator, the swirling of the fluid is provided by the guide vane mounted on the top of simulator and the circulation of the fluid in tornadoes could also be modelled. Most recently, Refan et al. (2014) developed a tornado simulator in WindEEE. In this simulator, the swirling of the fluid was provided the through the fans whose direction could be controlled. Fig. 2 summarizes the reproduced different types of tornadoes in different types of tornado simulators. In the studies listed in Fig. 2, the definition of the swirl ratio is same, S ¼ roГ /2Qh, where Г is the circulation magnitude, ro the radius of the updraft hole and Q represents the flow rate. Generally speaking, based on the definition of swirl ratio adopted, the single celled vortex occurs in the range of 0 < S < 0.4, and the two celled or multi celled vortex occurs in the range of S > 0.7. In the range of 0.4  S  0.7, the vortex is in a transient stage where the vortex breakdown appears. Focusing on the vortex breakdown, some interesting features have been found. In the following, a brief summary of the properties of the

* Corresponding author. E-mail address: [email protected] (Z. Liu). https://doi.org/10.1016/j.jweia.2018.04.009 Received 2 September 2017; Received in revised form 28 December 2017; Accepted 9 April 2018 Available online 8 May 2018 0167-6105/© 2018 Elsevier Ltd. All rights reserved.

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Journal of Wind Engineering & Industrial Aerodynamics 177 (2018) 306–326

Nomenclature A A Ari Azi Cr Cs D D Di h k Ls ~p P Pr Pr Pz Q r0 rc Re rs rt ~ij S T Ti ~i u

(U,V,W) uiuj Urs uτ Vc

aspect ratio advection budget of TKE radial advection term vertical advection term centrifugal force term Smagorinsky constant distance to the closest wall dissipation of TKE diffusion term height of the inlet layer [m] turbulence kinetic energy [m2s2] mixing length for subgrid-scales[m] filtered pressure[N m2] production of TKE radial pressure gradient term pressure diffusion of TKE axial pressure gradient term flow rate [m3 s1] radius of the updraft hole[m] Radius at which Vc occurs [m] Reynolds number radius of the convergent region[m] radius of the exhaust outlet[m] rate-of-strain tensor [s1] turbulence transportation of TKE turbulent force term filtered velocities [ms1]

Vrs W0 (x, y, z)

Mean radial, tangential and vertical velocities [m s1] Reynolds stresses [m2 s2] radial velocities at r ¼ rs[m s1] friction velocity[m s1] maximum mean tangential velocity at the cyclostrophic balance region[m s1] tangential velocity at r ¼ rs[m s1] velocity at the outlet[m s1] Cartesian coordinates[m]

Greek symbols δij Kronecker delta Г∞ free-stream circulation at the outer edge of the convergence region[m3 s1]  θ inflow angle [ ] κ von Karman constant μ Viscosity [Pa⋅s] μt SGS turbulent viscosity [Pa⋅s] ρ density [kg m3] τij SGS stress [N m2] Abbreviations DPM Discrete Phase Method LES large-eddy simulation MEM Maximum Entropy Method r.m.s root mean square SGS subgrid-scale SIMPLE semi-implicit pressure linked equations TKE turbulence kinetic energy

fields in the tornado-like vortex, more details of the flow can be provided. Refan and Hangan (2016) numerically studied tornadic vortices and concluded that vortex breakdown involves a complex interaction between two competing tendencies. The first is the production of an axial up-draught as an extension of the convergent boundary layer, and the second is the decay of the tangential velocity with height. Lewellen D.C. and Lewellen W.S. (2007) examined the near surface intensification of tornado-like vortices and showed that the vortex breakdown stage represents a sharp transition from states with strong upward axial flows to those with significantly large core radii, reduced axial velocities, and increased turbulence levels. In systematic research of examining four typical tornados by Liu and Ishihara (2015), the turbulence characteristics and force balances in the radial as well as vertical directions were evaluated, in which an extraordinarily high turbulence was found at the stage of sub-critical vortex breakdown (S ¼ 0.6). It is this case that will be examined in detailed in the present study as illustrated by red dashed line in Fig. 2. As found in the studies summarized above, at the sub-critical vortex breakdown stage, the turbulence in the fluid is very high. However, up to now no detailed explanation of this phenomenon was provided due to the insufficient recorded data. But the clarification of the high turbulence is a very important issue for wind resistant design when a structure is exposed to a tornado-like vortex sub-critical vortex breakdown stage. In this study, the mean and fluctuating flow fields, the force balance and the dynamics of the tornado at the vortex breakdown stage will be examined in detail using three-dimensional large eddy simulations, and the reasons for the occurrence of high turbulence at this stage will be revealed in a step-by-step manner. The correlation of the fluctuating parameters and the balance of turbulence kinetic energy (TKE) for subcritical vortex breakdown stage will be examined as well, which is important for understanding the structure of the turbulence. Spectrum analysis of the wind speeds for sub-critical vortex breakdown stage will be conducted, which is important for understanding the dynamic

tornado-like vortex at breakdown stage will be provided. The vortex breakdown (Harvey, 1962; Benjamin, 1962; Lugt, 1989) is considered to be an axisymmetric analog to the hydraulic jump phenomenon observed in channel flows. As S is increased, the altitude of the vortex breakdown decreases; until the breakdown is just above the surface (Church, 1979; Church and Snow, 1977). This state has been referred to as a ‘‘drowned vortex jump’’ (DVJ; Maxworthy, 1973) and is generally associated with having the highest near surface azimuthal wind velocities. When S is further increased, the vortex breakdown reaches the surface. The decrease of the altitude of the vortex bubble as increasing the swirl ratio is illustrated in Fig. 3. As observed in experiments by Refan et al. (2014), at the vortex breakdown stage the flow is highly unstable as it consists of three distinct dynamic regions: turbulent sub-critical region aloft followed by the breakdown bubble in the middle and the narrow super-critical core close to the ground. Following the discussion by Refan and Hangan (2016) at the stage where the vortex breakdown reaches the surface is considered to be “subcritical”. In that study they also found that the development of the free stagnation point towards the surface continues until it touches the ground at around S ¼ 0.57. Pauley (2010) performed pressure measurements of the vortex axis using a Ward-type tornado simulator, and the static pressure measured along the vortex axis increased with the height downstream of the vortex breakdown flow. In that study, force balance analysis in the axial direction was also performed, and it was concluded that turbulent stresses can play an important role in the axial momentum balance. From the experiment by Tari et al. (2010), we can see at the sub-critical vortex breakdown stage (S ¼ 0.68), the fluctuations of the velocity are very high. Due to the instabilities associated with the vortex break-down bubble and the transition from laminar to turbulent flow, one can expect considerable variations in the vortex characteristics and structures. Numerical simulation is another important way to study the transition stage–vortex breakdown. Using complete information for the flow 307

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Fig. 1. Illustrations of four of the stages as the swirl ratio is increased from zero: (a) single celled vortex; (b) surface vortex with vortex breakdown above the surface; (c) two celled vortex; and (d) multi celled vortex with stagnant core.

Fig. 2. Summary of the modelled tornado-like vortices as a function of swirl ratio, the red dashed line indicates the case examined in the present study. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

fields will be provided, including the mean and fluctuations of the flow, as well as the correlations of the flow parameters. The force balances in the radial and vertical directions, and the kinetic energy balance of the turbulence will also be introduced in section 3 to elucidate the dynamic

performance as well as the structure dynamic response to the tornado. In section 2, the details of the model are introduced, including the dimensions, grid distribution and boundary conditions. In section 3, detailed information regarding the three-dimensional turbulent flow

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Fig. 3. Illustrations of decreasing of the altitude of the vortex bubble as the swirl ratio is increased: (a,b) surface vortex with vortex breakdown above the surface; (c) drowned vortex jump (breakdown just above the surface, sub-critical vortex breakdown stage).

follows:

performance of the vortex, followed by an examination of the time histories of the wind speeds, including a spectrum analysis as well as flow visualizations in section 4.

1 3

τij ¼ 2μt ~Sij þ τkk δij ;

2. Numerical model and boundary conditions

 





(4)

in which Ls denotes the mixing length for subgrid-scales, κis the von Karman constant, i.e., 0.42, d is the distance to the closest wall and V is the volume of a computational cell. In this study, Cs is the Smagorinsky constant, which is determined to be 0.032 based on Oka and Ishihara (2009).

Momentum and mass are primarily transported by large eddies; therefore, a large-eddy simulation (LES) is adopted to simulate tornadolike vortices. In such simulations, large eddies are directly computed, while the influence of eddies smaller than the grid spacing are parameterized. The Boussinesq hypothesis is employed, and the standard Smagorinsky-Lilly model (Smagorinsky, 1963; Lilly, 1992) is used to calculate subgrid-scale (SGS) stresses. The governing equations are obtained by filtering the time-dependent Navier-Stokes equations in Cartesian coordinates (x, y, z), which can be expressed as follows:

2.2. Configurations of the numerical tornado simulator 2.2.1. Computational domain and mesh A Ward-type simulator is chosen, which is geometrically the same as that in the study by Matsui and Tamura (2009); the configurations of the numerical model are shown in Fig. 4(a). The two most important geometrical parameters are the height of the inlet layer, h, and the radius of the updraft hole, r0, which are set at 200 mm and 150 mm, respectively. The dominant parameter determining the structure of the tornado vortex is identified as the swirl ratio, which will be discussed in section 2.3. The flow rate is calculated by Q ¼ π rt2 W0 ¼ 0:3m3 s1 , where rt ¼ 250 mm is the radius of the exhaust outlet and W0 ¼ 9.55 m s1 is the velocity at the outlet. The Reynolds number is expressed as Re ¼ 2r0W0/υ ¼ 1.60  105 following the study by Monji (1985).

(1)


qffiffiffiffiffiffiffiffiffiffiffiffi

μt ¼ ρL2s ~S ¼ ρLs 2~Sij ~Sij ; Ls ¼ min κd; Cs V 1=3

2.1. Governing equations



∂ρ~ui ∂ρ~ui ~uj ∂ ∂~u ∂~p ∂τ þ ¼ μ i   ij ∂t ∂xj ∂xj ∂xi ∂xj ∂xj

(3)

~ij is the rate-of-strain where μt denotes the SGS turbulent viscosity, S tensor for the resolved scale, and δij is the Kronecker delta. The Smagorinsky-Lilly model is used for the SGS turbulent viscosity:

The governing equations and configurations for the numerical tornado simulator are introduced in section 2.1 and section 2.2, followed by the introduction of the swirl ratio applied in this study.

∂ρ~ui ¼0 ∂xi


 ~Sij ¼ 1 ∂~ui þ ∂~uj 2 ∂xj ∂xi

(2)

~i and~p are the filtered velocities and pressure, respectively, μ is where u the viscosity, ρ is the density, and τij is the SGS stress, which is modeled as

Fig. 4. Configuration of the computational domain (a) and the mesh system (b). 309

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height z1 are set at 0.24 m s1 and 0.01 m, respectively, by matching the velocity profile at the inner ring of the guide vanes as described by Ishihara et al. (2011), and θ is the inflow angle. In Eq. (7), U1 is a constant; moreover, Urs is a function of z. The inflow angle θ is adjusted to change the swirl ratio. Porous medium is applied to model a honeycomb in which no drag force is added in the vertical momentum equation but almost infinite drag forces are added in the horizontal directions. Therefore, the fluid can move freely in the vertical direction with nearly no motion in the horizontal directions, which is similar to the fluid in a honeycomb. At the outlet, the outflow boundary condition is used, which means that the gradients in pressure and velocities are set to zero herein. The boundary conditions are summarized in Table 3.

Considering the axisymmetrical nature of tornado-like vortices, an axisymmetric grid is adopted, shown in Fig. 4(b). With the intent to quantitatively investigate the turbulent features in the vicinity of the center and the region near the ground, a fine mesh is considered in the convergence region, in which 86 nodes in the radial direction and 45 nodes in the vertical direction are used; the minimum size of the mesh is approximately 1 mm in the radial direction and 0.1 mm in the vertical direction. The spacing ratios in the two directions are less than 1.2 to avoid a sudden change in the grid size. The total number of grid points is approximately 7.8  105. We have compared the simulated flow field with that by Ishihara and Liu (2014), in which the grids are coarser than in the present study. The numerical results for the two simulations have shown good agreement with the experiment reported by Matsui and Tamura (2009), indicating that the results are grid-independent. The geometry parameters of the computational domain and the mesh are summarized in Tables 1 and 2.

2.2.3. Solution scheme and solution procedure The finite volume method (FVM) is used for the simulations in which the variables are distributed in a non-staggered, cell-centered mesh system. The second-order central difference scheme is used for the convective and viscous terms, and the second-order implicit scheme is employed for the unsteady term:

2.2.2. Boundary conditions When the wall-adjacent cells are in the laminar sublayer, the wall shear stresses are obtained from the following laminar stress-strain relationship: ~ ρuτ y u ¼ uτ μ


 dϕ 3ϕn  4ϕn1 þ ϕn2 ¼ ; Δtn ¼ t  tn1 ¼ tn1  tn2 dt n 2Δtn

(5)

where indices “n” and “n-1” denote the new and old time levels, respectively. Time step size Δt is set as 0.0001s and in convective time units Δt* ¼ ΔtW0/r0 ¼ 0.0063. The solution method consists of linearization of the non-linear equations and implementation into a matrix solution. The predicted conjugate gradient (PCG) method is applied to solve the linearized equations along with Algebraic Multi-Grid (AMG) approach. The Courant Friedrichs Lewy (CFL) (Courant et al., 1928) number is based on the time step size(Δt), velocity(ui), and grid size(Δxi), expressed as C ¼ ΔtΣui/Δxi. Here the CFL number is limited to not exceed 1, Cmax ¼ 1, in the whole computational domain. The SIMPLE (semi-implicit pressure linked equations) algorithm is used to solve the discretized equations, which was introduced by Ferziger and Peric (2002). Relaxation factors were employed to promote the stability of the process. These relaxation factors are 0.3 and 0.7 for pressure and momentum respectively. Commercial software Ansys Fluent 14.5 (Ansys Inc, 2014) is used for the calculations. The numerical schemes are summarized in Table 4. The initial transient effects are found to disappear after 10 s;

If the mesh cannot resolve the laminar sublayer, it is assumed that the centroid of the wall-adjacent cells falls within the logarithmic region of the boundary layer, and the law-of-the-wall is employed as follows:
 ~ u 1 ρuτ y ¼ ln E uτ κ μ

(6)

~ is the filtered velocity tangential to the wall, y is the distance where u between the center of the cell and the wall, uτ is the friction velocity, and the constant E is 9.793. In tornado-like vortices, a flow with both axial and radial pressure gradients is present; however, the radial pressure gradient dominates the axial pressure gradient in the near-surface region, which implies the wall function can be used. In most of the region, the wall-adjacent cells are in the laminar sub-layer. At most regions of the computational domain, yþ is below 5, and during the calculations the maximum instantaneous yþ is below 6. The velocity profiles at the inlet are specified as follows: 8
1=n > : Vrs ¼ Urs tanðθÞ

Table 3 Summary of the boundary conditions.

(7)

where Urs and Vrs are radial velocities and the tangential velocities at r ¼ rs (see Fig. 4), n is equal to 7, the reference velocity U1 and reference

Locations

Boundary type

Expression

Outlet of the simulator Surrounding walls of the simulator Inlet of the chamber

Outlet Non-slip wall

∂ui =∂n ¼ 0, ∂p=∂n ¼ 0 ∂p=∂n ¼ 0, ui ¼ 0

Velocity inlet

ui determined by Eq. (7), ∂p=∂n ¼ 0

Table 1 Summary of geometrical parameters of the computational domain. Radius of convergent chamber (rs) (m)

Height of convergent chamber (h) (m)

Radius of updraft hole (r0) (m)

Height of convective chamber (l) (m)

Radius of convective chamber (rm) (m)

Radius of outlet (rt) (m)

1.0

0.2

0.15

0.6

0.6

0.125

Table 4 Summary of numerical schemes.

Table 2 Summary of grid parameters of the numerical model. Directions

Minimum grid size (m)

Maximum grid size (m)

Maximum grid increasing ratio α

Grid number N

Vertical Tangential Radial

0.1 1.0 1.0

5 15 10

1.2 1.2 1.2

7.8  105

(8)

Time discretization scheme

Second-order implicit scheme

Cs number

0.032

Space discretization scheme

FVM second-order central difference scheme 0.0063

SGS model

SmagorinskyLilly

CFL number: ΔtΣui/Δxi

rc, the vertical advection term approaches zero, while the centrifugal term balances only with the pressure gradient term. However, after moving to a higher elevation of z ¼ 0.2 h, the radial advection term emerges, and the peak of the turbulence term occurs at r ¼ 1.0rc. Comparison of Fig. 12(a) and (b) shows that the centrifugal force and pressure gradient force are much larger at z ¼ 0.1 h than those at z ¼ 0.2 h because z ¼ 0.1 h is the location just below the expansion of the core. As a result, the tangential momentum can move closer to the center, resulting in a very large centrifugal force. With an increasing height, the balance mainly occurs in between the pressure gradient force and the centrifugal force. The magnitudes of the forces at elevations higher than 0.4 h, as shown in Fig. 12(d)~Fig. 12(f), are almost equivalent, indicating that a cyclostrophic balance has been achieved. Further examination of the sub-terms of the turbulence force in the radial balance is also carried out, as shown in Fig. 13, where Tu1 indicates the radial gradient of the Reynolds normal stress u2, Tu2 the vertical gradient of uw, Tu3 –v2/r, and Tu4 u2/r. Fig. 13 clearly demonstrates that the sub-terms show large values at z ¼ 0.1 h and that the largest sub-terms are always –v2/r and u2/r which are close to the center regardless of the elevation. It is interesting that –v2/r and u2/r have almost the same magnitude, so that even if their magnitude is large they can cancel out one another, indicating that the source of the normal Reynolds stresses u2 and v2 should be the same, which will be clarified in a later discussion. The vertical balance of the Reynolds averaged Navier-Stokes equation in the cylindrical coordinate can be expressed as follows:

moves from negative to positive, the horizontal fluctuations increase. Another interesting finding is obtained from w’ and p’ at Point 2. When the magnitude of the negative vertical fluctuation is over 0.5Vc, the pressure fluctuation becomes almost a constant. In contrast, when the vertical fluctuation is positive, the scatters are almost located on a straight line, and the slope is the same as that at Point 1. When increasing the height to Point 3, it shows a greater concentration of the scatters of v’ vs u’, w’ vs u’, and w’ vs v’, but the vertical fluctuation is mainly scattered at w’ ¼ -1Vc with an absolute value which is slightly smaller than that at Point 2. For w’ vs p’, the horizontal correlation becomes weak and less scattered, with a large negative pressure fluctuation. As the height increases from Point 4 to Point 6, the above trend persists. 3.3. Reynolds averaged momentum balance In a previous study by Ishihara et al. (2011), the vertical force balance in a tornado-like vortex with a low swirl ratio and radial force balance for a high swirl ratio using the Reynolds averaged axisymmetric Navier-Stokes equations was investigated. Liu and Ishihara (2015) carried out studies systematically investigating the force balance of four typical tornadoes. However, the examinations mainly focused on the locations at the center of the tornado or the location very close to the ground. The sub-terms of the turbulence force in the vertical balance has never been studied. The radial balance of the Reynolds averaged Navier-Stokes equation in the cylindrical coordinate can be expressed as follows:


U



∂U ∂U V 2 1 ∂P ∂u2 ∂uw v2 u2 þW  ¼  þ  þ þ Du ∂r ∂z r ρ ∂r ∂r ∂z r r

(10)

The left-hand side consists of the radial advection term Aru, the vertical advection term Azu, and the centrifugal force term Cr. The right-hand side of the equation includes the radial pressure gradient term Pr, turbulent force term Tu, and diffusion term Du. The diffusion term Du in this equation is sufficiently small to be ignored compared with the other terms. r is the radial distance away from center (x ¼ 0, y ¼ 0).




U

315



∂W ∂W 1 ∂P ∂uw ∂w2 uw þ Dw : þW ¼  þ þ r ∂r ∂z ρ ∂z ∂r ∂z

(11)

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Journal of Wind Engineering & Industrial Aerodynamics 177 (2018) 306–326

Fig. 11. Scatters of the fluctuations of v’ vs u’, w’ vs u’, w’ vs v’, and w’ vs p’.

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10

2

5

Aru Azu Cr Pr Tu

b.

Aru Azu Cr Pr Tu

Radial momentum budget

Radial momentum budget

a.

0

-5

1

0

-1

z=0.1h 0

1

2

r/r

3

4

5

0

1

2

c

2

1

r/r

3

-1

Aru Azu Cr Pr Tu

1

0

-1

z=0.3h 1

2

r/r

3

4

5

0

1

2

c

2

1

r/r

3

-1

Aru Azu Cr Pr Tu

1

0

-1 z=0.6h

z=0.5h 0

1

2

r/r

3

4

5

c

e.

0

-2

4

2

Aru Azu Cr Pr Tu

e. Radial momentum budget

z=0.4h -2

Radial momentum budget

0

5

d.

0

-2

4

c

2

Aru Azu Cr Pr Tu

c. Radial momentum budget

z=0.2h -2

Radial momentum budget

-10

5

c

-2

0

1

2

r/r

3

4

5

c

Fig. 12. Reynolds averaged force balance in radial direction at heights of (a) z ¼ 0.1 h, (b) z ¼ 0.2 h, (c) z ¼ 0.3 h, (d) z ¼ 0.4 h, (e) z ¼ 0.5 h and (f) z ¼ 0.6 h.

around z ¼ 0.2 h. From z ¼ 0.2 h to z ¼ 0.3 h, the main balance occurs between the turbulence and the pressure gradient. Above z ¼ 0.3 h, all the terms become almost zero. Moving the radial location to r ¼ 0.5rc, as shown in Fig. 14(b), we find that the turbulence force term shows a similar trend to that at r ¼ 0.5rc, increasing from zero to a high value and then changing to a negative value. However, the vertical locations of the peak positive and negative force terms shift upward. In addition, the radial advection term emerges, exhibiting the same order of magnitude as the vertical advection term. In the region close to the ground, z < 0.1 h, the advection terms (radial advection term and vertical advection term) are mainly balanced with the pressure gradient term. Due to the presence of the funnel shape of the vortex bubble, with an increasing radial distance, the peaks of the turbulence term move further upward, as can be observed in Fig. 14(c) and (d). Further examination of the sub-terms of the turbulence force in the

The left-hand side consists of the radial advection term Arw and the vertical advection term Azw. The right-hand side represents the axial pressure gradient term Pz, turbulent force term Tw, and diffusion term Dw. The diffusion term in the equation is sufficient to be ignored in comparison with the other terms. Four radial locations, r ¼ 0.0rc, r ¼ 0.5rc, r ¼ 1.0rc, and r ¼ 2.0rc, are chosen to draw the balances, as shown in Fig. 14, in which all of the terms are normalized to V2c . The terms in the vertical balance show the largest values at the center of the tornado, as plotted in Fig. 14(a). Close to the ground, at z < 0.08 h, the balance occurs mainly between the vertical advection and the vertical pressure gradient. The radial advection term provides a value of zero at the center due to the axis-symmetry of the time-averaged flow fields. At the vertical location around z ¼ 0.1 h, the turbulence force shows a peak positive value. With increasing elevation, the turbulence force changes to a negative value and reaches a peak negative value at the vertical location

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10

2

Tu Tu1 Tu2 Tu3 Tu4

5

Tu Tu1 Tu2 Tu3 Tu4

b. Turbulent sub-terms

Turbulent sub-terms

a.

0

-5

1

0

-1

z=0.1h 0

1

2

r/r

3

4

5

0

1

2

c

2

1

r/r

3

-1

Tu Tu1 Tu2 Tu3 Tu4

1

0

-1 z=0.4h

z=0.3h 1

2

r/r

3

4

5

0

1

2

c

1

r/r

3

4

5

c

2

Tu Tu1 Tu2 Tu3 Tu4

e. Turbulent sub-terms

-2

Tu Tu1 Tu2 Tu3 Tu4

f. Turbulent sub-terms

0

2

0

-1

1

0

-1 z=0.6h

z=0.5h -2

5

d.

0

-2

4

c

2

Tu Tu1 Tu2 Tu3 Tu4

c. Turbulent sub-terms

z=0.2h -2

Turbulent sub-terms

-10

0

1

2

r/r

3

4

5

c

-2

0

1

2

r/r

3

4

5

c

Fig. 13. Radial distribution of the sub-terms in radial turbulence forces at heights of (a) z ¼ 0.1 h, (b) z ¼ 0.2 h, (c) z ¼ 0.3 h, (d) z ¼ 0.4 h, (e) z ¼ 0.5 h and (f) z ¼ 0.6 h.

the balance of TKE is still not clear. Production of TKE is mainly determined by the shear of the mean flow; additionally, the production of kinetic energy working against the Reynolds stresses removes kinetic energy from the mean flow and transfers it to the fluctuating velocity fields. Advection budget represents the transportation of TKE by the mean velocities. Pressure diffusion budget determines the turbulence transportation of TKE by the pressure fluctuations. Turbulence transportation budget is the turbulence transportation of TKE by the velocity fluctuations. And dissipation budget represents the sink of the turbulent kinetic energy budget working against the fluctuating deviatoric stresses and transforming kinetic energy into internal energy. From the examination of the TKE budget, the contribution of the TKE from the production, advection, turbulent transport, and the dissipation could be clarified. Eq. (11) shows the TKE budget formulation:

vertical balance is also carried out as shown in Fig. 15, where Tw1 indicates the radial gradient of the Reynolds shear stress uw, Tw2 the vertical gradient of w2 and Tw3 uw/r. At r ¼ 0.0rc, the total turbulence force is mainly derived from ∂w2/∂r, and their trends as well as the reversing points are almost equivalent. In addition, the effects of Tw1 consistently cancel out the total turbulence force. 3.4. Turbulence kinetic energy balance The production (P) of turbulence kinetic energy (TKE) in the tornadolike vortex was studied by Tari et al. (2010) using a tornado simulator in the laboratory. However in that study, the other terms for the TKE budget, such as the advection (A), pressure diffusion (Pr), turbulence transportation (T), and dissipation (D), are not examined, and therefore 318

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Fig. 15. Vertical distribution of the sub-terms in vertical turbulence forces at radial distances of (a) 0rc, (b) 0.5rc, (c) 1.0rc and (d) 2.0rc. 319

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Uj

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∂k 1 ∂ui p 1 ∂uj uj ui ∂ Ui ∂ui ∂ui ¼   ui uj ν ∂xj ρ ∂xi 2 ∂xi ∂xj ∂xj xj

corresponding pressure diffusion (also called the gradient of the Reynolds stress flux due to fluctuating pressure) of the Reynolds normal stresses can be evaluated. At places near the vortex bubble, the pressure diffusion is significant, and its magnitude is much larger than those of the turbulence transportation and production terms, indicating the pressurevelocity correlations and subsequently the pressure diffusions have substantial impacts on the dynamics of turbulence transport in the vortex bubble. In contrast, the pressure diffusion decreases quickly away from the bubble and is negligible at elevations above z ¼ 0.4 h. Moreover, the distribution patterns between the pressure diffusion and the advection terms are very similar, but with opposite signs, above the vortex bubble. This similarity between the two distribution patterns is in agreement with the general understanding of the diffusion mechanism, i.e., newly

(12)

in which, the advection term A is -Uj⋅∂k/∂xj, pressure diffusion term Pr is 1/ρ⋅∂uip/∂xi, turbulence transportation T is 1/2⋅∂ujujui/∂xi, turbulence production P is -uiuj⋅∂Ui/∂xj, and dissipation term D is -ν⋅∂ui/∂xj⋅∂ui/∂xj. The budget is presented in Fig. 16 at six elevations which cross the six monitoring points introduced previously. The advection term A has the effect of sinking TKE at z ¼ 0.1 h near the center of the tornado, as shown in Fig. 16(a), but it quickly decreases to zero when r > 0.5rc. The sinking effects of the advection term are concentrated in the near surface region, which changes to a positive value from z ¼ 0.2 h. Based on the distribution of the pressure-velocity correlations provided above, the

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of the mean flow into TKE. Additionally, based on the formulation of the turbulence production term, the value is mainly determined by the shear of the mean flow. As a result, the greatest production term is clearly close to the boundary of the vortex bubble.

generated turbulence diffuses to places lacking a concentration of newly generated turbulence. The positive turbulence transportation at the vortex bubble represents the transportation of turbulence fluctuation energy away from the maximum mean shear location. The production of TKE means the net conversion rate per unit mass from the kinetic energy

Fig. 17. Time histories of the radial velocities at (a) Point 1, (b) Point 2, (c) Point 3, (d) Point 4, (e) Point 5, and (f) Point 6.

Fig. 18. Time histories of the vertical velocities at (a) Point 1, (b) Point 2, (c) Point 3, (d) Point 4, (e) Point 5, and (f) Point 6. 321

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signal at z ¼ 0.2 h which becomes weak at z ¼ 0.4 h. The time histories of the vertical velocity show quite different characteristics. First, at all heights, the low frequency motion is not obvious. In addition, the fluctuation of the vertical velocity is mainly derived from the moderate frequency motion. Finally, this moderate frequency motion shows the largest fluctuation at z ¼ 0.2 h. To further study the motions in the tornado at the sub-critical vortex breakdown stage, the power spectra of the velocity fluctuations u’ and w' are calculated by the Maximum Entropy Method (MEM). Fig. 19 shows the normalized spectrum of the radial velocity at six monitoring points, where the frequency is normalized to Vc =2π rc and the spectrum is normalized to r.m.s.2/n. In this way, one can clearly compare the frequencies with that of the tornado core rotation. At z ¼ 0.1 h, three peaks could be identified at 0.011, 0.08 and 0.95, corresponding to the abovediscussed low frequency, moderate frequency and tornado core rotation, respectively, in which the low frequency energy dominates. The spectrum of low frequency motion then becomes weak with increasing height, and the maximum spectrum of moderate frequency motion occurs at z ¼ 0.2 h. At z > 0.4 h, this moderate frequency motion almost disappears, while the core rotation energy of the tornado becomes the major portion. The normalized spectrum of the vertical velocity is shown in Fig. 20, where the motion energy is concentrated in the range with a normalized frequency which is equal to approximately 0.1 when z < 0.5 h, implying that the fluctuating vertical velocity is mainly attributed to a type of motion with a moderate frequency. Even though three types of motion have been identified, the

4. Instantaneous flow fields In the above discussion about the flow field statistics, several interesting points have been found, such as the extraordinarily large vertical velocity fluctuations, the differences in elevations of peak fluctuations, the unique distribution of the fluctuation scatters and the cancelling out of large sub-terms of turbulence force in the radial direction. With the intent to provide explanations for the above phenomenon, the instantaneous flow fields are studied, including the time histories at the monitoring points, the spectrum analysis of the instantaneous velocities, and the snapshots of instantaneous flow fields using flow visualization techniques. Time histories of the radial and vertical velocities at the six monitoring points, (x ¼ 0, z ¼ 0.1 h), (x ¼ 0, z ¼ 0.2 h), (x ¼ 0, z ¼ 0.3 h), (x ¼ 0, z ¼ 0.4 h), (x ¼ 0, z ¼ 0.5 h), and (x ¼ 0, z ¼ 0.6 h), are drawn in Fig. 17 and Fig. 18, where the horizontal axis is normalized to 2π rc =Vc which is the period of one tornado core rotation. From Fig. 17, there is an obvious low frequency motion at z ¼ 0.1 h. According to the normal Reynolds stress u at z ¼ 0.1 h and assuming this low frequency motion has a period of 240π rc =Vc , a sinusoidal curve is drawn which approximately fits the time history of the radial velocity. The sinuous shape of the radial velocity series yields a zero time-averaged value and large Reynolds normal stress u, which is similar to the results derived from an organized swirl motion in the study by Ishihara and Liu (2014). With increasing height, the strength of this low frequency motion becomes low, and another moderate frequency motion emerges with a marked

Fig. 19. Spectrum of radial velocity at (a) Point 1, (b) Point 2, (c) Point 3, (d) Point 4, (e) Point 5, and (f) Point 6. 322

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Fig. 20. Spectrum of vertical velocity at (a) Point 1, (b) Point 2, (c) Point 3, (d) Point 4, (e) Point 5, and (f) Point 6.

overshooting, and the small pressure drop corresponds to the dead flow in the core of tornado. The large pressure fluctuation at the instantaneous center may indicate a vertical vibration of the bubble. To show the moderate frequency motion clearly, the Discrete Phase Method (DPM) is applied to model the smoke in the experiment, in which tracer particles are injected from the bottom of the model and selected time points are illustrated in Fig. 18(b). The diameter of the tracer particles is uniform and equal to 1  105 m, and the injection rate is 0.1 g s1. The gravity of the particles is neglected, and their positions are directly integrated without considering their interaction with the flow. The particles are not released until the flow fields are in the quasi-steady stage to eliminate the effect of initial transients in the solution. The vortex bubble is also not stable in the vertical direction. It vibrates with time around the mean stagnation point (ze0.15 h). It is known that the vertical velocity in the bubble is nearly zero, but there is a large acceleration of the vertical velocity from the bottom to the stagnation point. Therefore, due to the vertical vibration of the vortex bubble, the mean stagnation point will experience a large vertical velocity and almost zero vertical velocity periodically with a moderate frequency. Thus, the largest w occurs at a location larger than those of u and v, and the fluctuation scatters u'w’ and v'w’ will exhibit unique distributions. When the vertical velocity fluctuation is positive, which means the vortex bubble moves upward, the flow can move closer to the center. Additionally, because of the above-described horizontally organized swirling motion, the tangential as well as the radial fluctuation will be very large at the

corresponding motion modes remain unclear. Therefore, four typical time steps are chosen, and the flow fields are visualized to understand the motion modes. For the low frequency motion, the selected four time points (t1, t2, t3, t4) are illustrated by red points in Fig. 21 and four time points (t5, t6, t7, t8) for moderate frequency motion are shown in Fig. 22. An instantaneous pressure horizontal distribution is plotted, as shown in Fig. 21 with superimposed vectors. An offset between the center of the mean flow fields (x ¼ 0, y ¼ 0) and those of the instantaneous flow fields (the location at which the horizontal velocity is zero) is obvious. This offset rotates around the point (x ¼ 0, y ¼ 0). When the value of u’ is large, v’ gives a small value, and vice versa, exhibiting a sinuous shape of the time histories. Therefore, the radial fluctuation and tangential fluctuation have the same amplitude but a π/2 phase difference. Considering that the phase difference has no effects on the calculation of the Reynolds stress, u and v should have the same values at (x ¼ 0, y ¼ 0) but corresponding mean velocities U and V should be zero. For this reason, the sub-terms Tu3 –v2/r, Tu4 u2/r cancel each other out in the radial force balance at the center, and there is a high probability of scatters of v’ vs u’ being located in a circle. At z ¼ 0.1 h, the offset is the largest between the instantaneous center and the mean center. With increasing height, this offset decreases, and at last these two centers coincide when z > 0.5 h. Consequently, at z ¼ 0.1 h, the time history of u' shows the largest amplitude and almost vanishes when z > 0.5 h. The pressure distribution shows an extraordinary fluctuation of the pressure drop at the instantaneous center. The large pressure drop corresponds to the vertical velocity 323

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Fig. 21. Instantaneous pressure distribution on the cross-sections of z ¼ 0.1 h, z ¼ 0.2 h, z ¼ 0.3 h, z ¼ 0.4 h, z ¼ 0.5 h, and z ¼ 0.6 h, at t1, t2, t3, and t4.

mean stagnation point. However, when the vertical velocity fluctuation is negative, which indicates the vortex bubble also moves downward, the mean stagnation point moves to the dead flow region, and the fluctuations u’, v’, and w’ are all very small. Therefore, a funnel shape of the u'w’ scatter and v'w’ scatter occurs. As indicated previously, the large pressure drop corresponds to the vertical velocity overshooting. Therefore, due to the vertical vibration of the bubble w'p’ scatter has the shape as shown in Fig. 11. When the pressure drop becomes large, which means the bubble moves upward, the vertical velocity fluctuation should be positive. Thus,

a negative correlation between w'p’ occurs when w’>0. 5. Conclusions In this study, a tornado was studied in detail at the sub-critical vortex breakdown stage using large eddy simulations. The averaged flow fields and fluctuations were examined, followed by correlations of the fluctuating velocities. The Reynolds averaged force balances in two directions were then examined. A detailed TKE budget examination of the tornado 324

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Fig. 22. . Instantaneous flow visualization by injecting tracer particles from the ground at (a) t5, (b) t6, (c) t7, and (d) t8.

 Spectrum analysis of the velocities was carried out, and three types of vibrations were identified. The instantaneous flow fields were visualized, and there is a clear offset between the center of the mean flow fields (x ¼ 0, y ¼ 0) and that of the instantaneous flow fields (the location at which the horizontal velocity is zero). This offset also rotates around the point (x ¼ 0, y ¼ 0). When u’ is assumed as large values, the value of v’ will be small, and vice versa, exhibiting a sinuous shape of the time histories. Therefore, the radial fluctuation and tangential fluctuation have the same amplitude. Consequently, the sub-terms–v2/r and u2/r cancel out one another in the radial force balance at the center, and there is a high probability that scatters of v’ vs u’ are located at the ring. At z ¼ 0.1 h, the offset between the instantaneous center and the mean center is the largest. With increasing height, this offset decreases, and these two centers finally coincide when z > 0.5 h.  The vortex bubble is also not stable in the vertical direction. It vibrates with time around the mean stagnation point (ze0.15 h). The vertical velocity is nearly zero in the bubble. However, there is a large acceleration in the vertical velocity from the bottom to the stagnation point. Therefore, due to the vertical vibration of the vortex bubble, the mean stagnation point will experience a large vertical velocity and almost zero vertical velocity periodically. When the vertical velocity fluctuation is positive, which means the vortex bubble moves upward, the high flow momentum will be transported closer to the center. Additionally, due to the horizontally organized swirling motion, the tangential as well as the radial fluctuation are very large at the mean stagnation point. However, when the vertical velocity fluctuation is negative, which means the vortex bubble moves downward, the mean stagnation point will be in the dead flow region and the fluctuations u’, v’, and w’ will be very small. Therefore, a funnel shape of u'w’ scatter and v'w’ scatter occurs.

flow fields was conducted for the first time. Time histories of wind velocities at some representative points were recorded, and to identify typical motions in the tornado, spectrum analysis was subsequently performed. Finally, flow field visualization was carried out to reveal the mechanism of the motions, and explanations of some interesting phenomena were provided. The findings of this study are summarized below.  The turbulence scatters were examined. At the point (z ¼ 0.1 h, r ¼ 0) for the scatters u’ vs v’, a high probability is located at the ring of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 u' þ v' ¼ 1:0Vc For w’ vs u’, w’ vs v’, the scatters form almost the same funnel shapes near the locations close to the mean stagnation point. They are concentrated when the vertical fluctuation is close to 2.0Vc. However, when the vertical fluctuation moves from negative to positive, the horizontal fluctuations increase.  The Reynolds averaged force balance was studied. Close to the ground, z < 0.08 h, the vertical balance is mainly in between the vertical advection and the vertical pressure gradient. For the radial balance, when z ¼ 0.1 h, the primary balance occurs between the centrifugal term, pressure gradient term, turbulent term and vertical advection term. With increasing height, the balance occurs mainly between the pressure gradient force and the centrifugal force, with almost equivalent magnitudes at elevations higher than 0.4 h, indicating a cyclostrophic balance. For the sub-terms of the turbulence force in the radial direction, –v2/r and u2/r consistently occur close to the center regardless of the height of the elevation. In addition, –v2/r and u2/r have almost the same magnitudes.  The TKE budget was examined. At places near the vortex bubble, pressure diffusion is significant, and its magnitude is much larger than those of turbulence transportation as well as production terms, indicating that the pressure-velocity correlations and subsequently the pressure diffusions have a substantial impact on the dynamics of turbulence transport in the vortex bubble. The positive turbulence transportation at the vortex bubble represents a transportation of turbulence fluctuation energy away from the maximum mean shear location.

Acknowledgement Supports from 325

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(2016YFE0127900, 2016YFC0800200, 2016YFC0802002), the National Natural Science Foundations of China (51608220, 51720105005), the Project of Innovation-driven Plan in Huazhong University of Science and Technology (2017KFYXJJ141), National Science Foundation (NSF-AGS1419614), are gratefully acknowledged.

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