Quarterly Journal of the Royal Meteorological Society
Q. J. R. Meteorol. Soc. (2016) DOI:10.1002/qj.2739
Gravity currents propagating into ambients with arbitrary shear and density stratification: vorticity-based modelling M. M. Nasr-Azadani and E. Meiburg* Department of Mechanical Engineering, University of California at Santa Barbara, CA, USA *Correspondence to: E. Meiburg, Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA. E-mail:
[email protected]
We develop a vorticity-based approach for modelling quasi-steady, supercritical gravity currents propagating into a finite-height channel with arbitrary density and velocity stratification. The model enforces the conservation of mass, horizontal and vertical momentum. In contrast to previous approaches, it does not rely on empirical, energybased closure assumptions. Instead, the effective energy loss of the flow can be calculated a posteriori. The present model results in the formulation of a second-order, nonlinear ordinary differential equation (ODE) that can be solved in a straightforward fashion to determine the gravity-current velocity, along with the downstream ambient velocity and density profiles. Comparisons between model predictions and direct numerical simulations (DNS) show excellent agreement. Furthermore, they indicate that, for high Reynolds numbers, the gravity-current height adjusts itself so as to maximize the loss of energy. Key Words:
gravity current; background shear; stratified flow; vorticity-based model
Received 19 June 2015; Revised 19 December 2015; Accepted 3 November 2015; Published online in Wiley Online Library
1. Introduction Gravity currents are driven by horizontal gradients in hydrostatic pressure as a result of density differences (Benjamin, 1968; Simpson, 1997; Linden, 2012). Depending on the source of these density differences, various types of gravity currents can be distinguished, among them sea breezes, thunderstorm outflows, turbidity currents, pyroclastic flows, intrusive flows or powdersnow avalanches (Moncrieff, 1978, 1992; Hopfinger, 1983; Xu and Moncrieff, 1994; Branney and Kokelaar, 2003; Mehta et al., 2002; Sutherland et al., 2004; Flynn and Linden, 2006; Meiburg and Kneller, 2010). In the atmosphere, gravity currents frequently interact with background shear and/or density stratification. For instance, severe thunderstorms can produce gravity currents of cold air travelling along the ground, the dynamics of which are affected by existing shear in the ambient atmosphere (Bryan and Rotunno, 2014a), thereby resulting in complex flow structures. Field observations and measurements suggest that such thunderstorm outflows and their nonlinear coupling with the ambient shear may produce long-lived squall lines (Moncrieff, 1978, 1992; Rotunno et al., 1988; Xu and Moncrieff, 1994; Weisman and Rotunno, 2004), along with associated strong vertical convection (Weckwerth and Wakimoto, 1992). The interaction of gravity currents with background shear has been the subject of several previous studies. Some of these aim to extend the basic gravity current model of Benjamin (1968) to idealized ambient conditions such as constant shear or discrete velocity jumps (Xu, 1992; Xu and Moncrieff, 1994; Liu and Moncrieff, 1996b; Xu et al., 1996; Xue et al., 1997; Xue, 2000,2002; Bryan and Rotunno, 2014a,2014b). Just like Benjamin’s original c 2016 Royal Meteorological Society
model, these extensions usually require an empirical closure assumption to determine the gravity-current velocity, which can result in uncertainty regarding the model predictions. On the other hand, Borden and Meiburg (2013a) recently introduced an alternative model for gravity currents propagating into quiescent, unstratified ambients that avoids the need for an empirical closure assumption by enforcing the conservation of both horizontal and vertical momentum, based on the vorticity equation. By comparing model predictions with direct numerical simulation (DNS) results, the authors demonstrate that their approach results in improved accuracy compared with earlier ones. Most recently, Nasr-Azadani and Meiburg (2015) extended this vorticity-based modelling approach to gravity currents propagating into unstratified ambients with constant or two-layer shear and demonstrated very good agreement with DNS results. Realistic atmospheric gravity current models furthermore need to account for the effects of density stratification, which can result in the formation of internal waves or bores, as well as the thickening or thinning of the gravity current (Long, 1953; Crook and Miller, 1985; Rottman and Simpson, 1989; Liu and Moncrieff, 1996a; Ungarish and Huppert, 2002; White and Helfrich, 2008, 2012; Tan et al., 2011; Flynn et al., 2012; Winters and Armi, 2014). Consequently, the present work aims to extend the vorticity-based modelling approach of Borden and Meiburg (2013a) to gravity currents interacting with arbitrary background shear and density stratification, without the need for empirical closure assumptions. While our model development focuses on stably stratified conditions and steady flows, we also employ DNS simulations to investigate the influence of unsteady effects, such as those caused by the production of interfacial Kelvin–Helmholtz instabilities.
M. M. Nasr-Azadani and E. Meiburg
Figure 1. Configuration of a gravity current (shown in gray) propagating into sheared and stably stratified background fluid. Upstream of the gravity current front, density and horizontal velocity profiles are given as general known functions ρi (y) and Ui (y).
Figure 2. Schematic of the defined control volume IJKL for the problem of a gravity current running into ambient fluid shown in Figure 1. Conservation of mass and vorticity are derived for this streamtube of width yi , for which the centre at the inlet is known (yi ), displaced at the outlet to an unknown height (yo ). See the text for discussion and derivations.
It is well known that the presence of ambient stratification may lead to the production of internal waves, which can potentially influence the dynamics of the gravity current (Ungarish and Huppert, 2002; Maxworthy et al., 2002; Ungarish, 2006; Birman et al., 2007; White and Helfrich, 2008, 2012). The experiments and simulations by Maxworthy et al. (2002) demonstrate that such internal waves affect the front velocity only for subcritical gravity currents, i.e. gravity currents with a front velocity smaller than the fastest internal wave velocity, due to energy transfer between waves and current (Ungarish and Huppert, 2006; White and Helfrich, 2008, 2012; Crosman and Horel, 2012; Lareau and Horel, 2015). Supercritical gravity currents, on the other hand, propagate in a quasi-steady fashion. The present investigation focuses on modelling such quasi-steady, supercritial gravity currents. Section 2 defines the problem and presents the model derivation. Section 3 presents the energy and head loss analyses, which provide insight into whether or not the model predictions are physically feasible. Section 4 validates the model by comparing its predictions with established results under certain limiting conditions. For a representative example involving complex ambient velocity and density profiles, section 5 demonstrates good agreement between model predictions and DNS simulation results. In addition, it shows that for high-Reynolds-number unsteady flows the gravity-current height adjusts itself to the value that corresponds to maximum energy dissipation. Section 6 summarizes the findings of the current study and presents the key conclusions. Appendix A extends the model framework to scenarios with discontinuous velocity and/or density profiles representing multi-layered structures in the ambient. 2. Problem definition and theoretical model We consider a gravity current of density ρ1 and given height h travelling from right to left with unknown constant velocity Ug in a horizontal tank of height H, cf. Figure 1. Note that the current height itself is usually determined by the far-field conditions of the flow, so that common models of the frontal region can determine the front velocity as a function of height, but not the height itself (Benjamin, 1968; Simpson, 1997; Linden, 2012; Borden and Meiburg, 2013a). The tank is filled with ambient fluid that has a known continuous density stratification ρi (y) and horizontal velocity Ui (y) far to the left of the current front. To ensure stably c 2016 Royal Meteorological Society
stratified conditions, we demand ρi (y) ≤ ρ1 , dρi ≤ 0. dy
(1) (2)
Without loss of generality, we can rewrite the density function as ρi (y) = ρa + δρi (y) ,
(3)
where ρa indicates the density at the top wall. In order to obtain a steady flow, we switch to the reference frame moving with the gravity-current velocity Ug . In this frame, the upstream velocity reads ui (y) = Ui (y) + Ug .
(4)
Let us now consider the narrow streamtube IJKL in Figure 2, with the dashed streamline at its centre. This streamline of width yi originates from a given height yi far upstream of the gravity current and is displaced vertically to yo far downstream of the current front. To describe the unique relationship between these two heights, we introduce yo = yi + ξ ,
(5)
where the vertical displacement ξ is only a function of the inlet height yi . By definition, no fluid crosses the two streamlines IJ and LK, so that the continuity equation takes the form (6) ui yi = uo yo . yi
yo
Here, yi and yo denote the upstream and downstream width of the streamtube respectively. Next, let us consider the vorticity balance for the streamtube, based on the two-dimensional, steady-state, inviscid vorticity equation in the Boussinesq approximation: u · ∇ω = −ˆg
∂ρ ∗ . ∂x
Here, gˆ and ρ ∗ represent the reduced gravity Q. J. R. Meteorol. Soc. (2016)
(7)
Gravity Currents in Sheared Stratified Ambients
gˆ = g
ρ1 − ρa , ρa
(8)
ρ − ρa , ρ1 − ρa
(9)
and dimensionless density ρ∗ =
respectively. Note that we do not assume the flow to be hydrostatic in the vicinity of the gravity current front. Furthermore, as a result of the Boussinesq approximation the pressure does not appear in the vorticity balance. By applying Gauss’s divergence theorem to the streamtube of width yi , Eq. (7) yields ∂ρ ∗ gˆ ωu · n d = − dA . (10) A ∂x Here, , A and n denote the boundary, area and unit outer normal vector of the streamtube of width yi , respectively. We assume that the density field is non-diffusive, so that the density is constant along each streamline. The integral describing the baroclinic vorticity production within the narrow streamtube can then be evaluated by approximating the continuous vertical density profile ρi (y) by a piecewise constant density profile with a density jump ρ(yi ) along the central streamline, where (11) ρ(yi ) = ρi yi − ρi yi , yi + 2
ρ(yi ) + uo ωo yo . ui ωi yi = −ˆg (yo − yi ) yi yo ρ1 − ρa
(12)
Dividing both sides of Eq. (12) by yi , applying the mass conservation Eq. (6) and letting yi → 0 yields (yo − yi ) dρi = ui ωo . ui ωi + gˆ (ρ1 − ρa ) dyi
(13)
Since the flow is horizontal far up- and downstream of the gravity current front, the vorticity values in Eq. (13) take the form =
ωo
=
dui , dyi duo − . dyo −
(14) (15)
Substituting Eqs (14) and (15) into Eq. (13) gives −ui
Along similar lines, we obtain for the vorticity at the outlet from Eq. (15) duo dyo duo 1 =− · dyi 1 + ξ 1 d ui · 1+ξ 1 =− · dyi 1 + ξ ui ξ − (1 + ξ )ui = . (1 + ξ )3
ωo (yo ) = −
(yo − yi ) dρi dui duo + gˆ = −ui . dyi (ρ1 − ρa ) dyi dyo
(16)
To simplify Eq. (16) further, we employ the streamline displacement ξ (see Eq. (5)) to relate the downstream (outlet) flow variables to their upstream (inlet) counterparts. By taking the derivative of Eq. (5) with respect to the inlet height yi , we obtain dyo = 1 + ξ , dyi
(17)
where ξ denotes the derivate of ξ with respect to yi . In the remainder, a prime indicates the derivative with respect to yi . Equation (17) provides us with the means to relate the derivate of any quantity (·) with respect to the outlet height to its derivative with respect to the inlet height via d(·) d(·) 1 = . dyo dyi 1 + ξ c 2016 Royal Meteorological Society
(18)
(20)
For a given streamline displacement function ξ , Eqs (19) and (20) provide the downstream velocity and vorticity profiles in terms of their upstream counterparts. By substituting Eqs (5) and (20) into the vorticity equation (16), we arrive at ξ u2i + ξ (1 + ξ )(2 + ξ )ui ui − ξ (1 + ξ )3
yi − 2
is the density difference across the streamtube of width yi . In this way, we obtain
ωi
In the limit as yi → 0, the mass conservation Eq. (6) gives dyo −1 uo (yo ) = ui (yi ) · dyi 1 = ui (yi ) · . (19) 1 + ξ
gˆ ρi = 0. (ρ1 − ρa ) (21)
Equation (21) represents a second-order nonlinear ordinary differential equation (ODE) for the vertical streamline displacement ξ as a function of the upstream velocity and density profiles. The required two boundary conditions are obtained from geometrical considerations, cf. Figure 2: AOA : ξ (yi = 0) = h ,
BB : ξ (yi = H) = 0 .
(22) (23)
Since, in the reference frame moving with the front, the gravitycurrent velocity Ug enters into the upstream velocity via Eq. (4), we require an additional equation to compute Ug . This can be obtained by integrating the vorticity equation (10) for a thin control volume that includes the vortex sheet separating the gravity current and the ambient fluid (OA in Figure 2) : u2o ρ1 − ρi (yi = 0) = gˆ h , 2 yo =h ρ1 − ρa ρ1 − ρi (0) . (24) uo (h) = 2ˆg h ρ1 − ρa Here, we have made use of the fact that the vorticity flux carried by a vortex sheet is given by the vortex sheet strength uo (h) multiplied by its principal velocity 0.5uo (h) (Saffman, 1992; Borden and Meiburg, 2013b). From Eq. (24), we obtain the downstream ambient fluid velocity uo (h) at A . Substituting Eqs (4) and (19) into Eq. (24) yields ui (0) − 1, uo (h) Ui (0) + Ug − 1. =
i (0) 2ˆg h · ρρ1 −ρ −ρ a 1
ξ (0) =
(25) (26)
Equation (26) provides us with an extra equation to obtain Ug , while solving Eq. (21) and its two boundary conditions (22) and (23) for ξ (yi ). Equation (21) cannot be solved analytically for general density and inflow velocity profiles. Consequently, we compute the gravity-current velocity Ug along with the displacement function ξ via an iterative numerical procedure in the following way. Q. J. R. Meteorol. Soc. (2016)
M. M. Nasr-Azadani and E. Meiburg (1) Guess the value of the gravity-current velocity Ug . (2) Compute ξ (0) from Eq. (26). (3) Solve Eq. (21) to compute ξ , based on the two initial conditions ξ (0) and ξ (0) (Eqs (22) and (26)). (4) Check whether or not the numerical value for ξ (H) satisfies the boundary condition (23). (5) Update Ug and repeat the loop until condition (4) is satisfied.
By substituting the upstream pressure profile (Eq. (30)) into Eq. (29), we obtain the upstream momentum integral as
The above derivation is valid for quasi-steady, supercritical gravity currents propagating into ambients characterized by continuous density and velocity profiles. Appendix A presents the extension to flows with discontinuous, multi-layered velocity and density profiles (Xue et al., 1997; Xue, 2000; White and Helfrich, 2012; Nasr-Azadani and Meiburg, 2015). To determine whether or not the gravity-current velocity Ug is supercritical, we define the Froude number
The downstream momentum integral is found in a corresponding fashion: H po + ρa u2o dy
Fr =
Ug . αN H
(27)
Here, N denotes the buoyancy frequency in the ambient, and α = 1/π for linearly stratified ambients. Supercritical currents are obtained for Fr > 1 (cf. Baines, 1984; Maxworthy et al., 2002; Munroe et al., 2009; White and Helfrich, 2012). 3. Energy loss and head loss analysis For given upstream velocity and density profiles, Eq. (21) allows us to compute the gravity-current velocity Ug as a function of the current height, along with the downstream density and velocity profiles. In order to check if the resulting flow is physically possible, we now analyze its energy and head loss. Note that, unlike Benjamin (1968) and Xu (1992), who enforce zero head loss along a specific streamline for closure, our assessment of the flow’s energy budget is performed a posteriori and solving for the gravity-current velocity Ug is not required. We begin by computing the head loss δBB along the top wall, cf. Figure 2. Here Bernoulli’s equation takes the form 1 1 = pB + ρa u2o + δBB . pB + ρa u2i yi =H yo =H 2 2
0
H
2
pi + ρa ui dy =
0
H
2
po + ρa uo dy .
(28)
pi (y) = pB + ρa g(H − y) +
0
pi + ρa u2i dyi
1 = pB + ρa gH 2 + 2
H
H
δρi g dη dyi . 0
(32)
yi
1 1 = pB + ρa gH 2 + (ρ1 − ρa )gh2 2 2 H H +h δρi g(1 + ξ ) dyi + ρa
u2i dyi 0 0 1+ξ H H + (1 + ξ ) δρi g(1 + ξ ) dη dyi . 0
(33)
yi
Note that in obtaining Eq. (33) we applied the following identities: dyo = dyi (1 + ξ ) , H H (·) dyo = (·)(1 + ξ ) dyi . h
(34) (35)
0
Equating the upstream and downstream momentum fluxes yields the pressure drop along the top wall: 1 1 pB − pB = − (ρ1 − ρa ) gh2 H 2 H +h δρi g(1 + ξ ) dyi 0 H
+
(1 + ξ )
H
−
H
δρi (1 + ξ )g dη dyi
yi
0
H
δρi g dη dyi
yi
− ρa 0
(29)
0
0
Here we assume stress-free conditions at the top and bottom walls. To integrate Eq. (29), we need to prescribe the pressure profiles far up- and downstream of the gravity current front. Here the flow is unidirectional, so that the pressure distribution across the channel height is hydrostatic. With the upstream density distribution (Eq. (3)), we obtain
H
Evaluating the head loss δBB requires information on the pressure drop p = pB − pB along the top wall. This can be obtained from the Boussinesq form of the horizontal momentum equation integrated across the entire channel:
H
ξ 2 u dyi . 1 + ξ i
(36)
Based on this pressure drop, we can now evaluate the head loss δBB along the top wall from Eq. (28). However, we note that for stratified ambients the head loss will vary across the streamlines, so that it is more informative to evaluate the energy loss in the streamwise direction integrated over the entire channel height. Towards that end, we write the energy equation for the entire control volume as E˙ i = E˙ o + δ E˙ .
(37)
H
δρi g dy .
(30)
y
Similarly, the downstream pressure distribution is evaluated as po (y) = pB + ρa g(H − y) ⎧ H ⎪ ⎨ ρ1 g(h − y) + h δρo g dy, + ⎪ ⎩ H δρo g dy, y
h ≥ y ≥ 0, H ≥ y ≥ h. (31)
c 2016 Royal Meteorological Society
Here, E˙ i , E˙ o and δ E˙ denote the up- and downstream energy fluxes as well as the energy loss across the gravity current front. For a specific flow to be physically feasible, we require δ E˙ ≥ 0. To ˙ we evaluate the up- and downstream energy integrals compute δ E, as H 1 E˙ i = ui pi + ρa u2i + ρi gy dy , (38) 2 0 H 1 2 ˙Eo = uo po + ρa uo + ρo gy dy . (39) 2 h Q. J. R. Meteorol. Soc. (2016)
Gravity Currents in Sheared Stratified Ambients Similarly to how we proceeded earlier for the momentum balance, the energy loss is thus obtained as δ E˙ = − pB − pB
H
+
1 − ρa 2
ui dyi +
0 H
ui 0
H
H
δρi gξ ui dyi 0
ξ δρi g dη dyi
Comparing Eq. (48) with Eq. (26), we find that the solution with the + sign in Eq. (48) is acceptable. By integrating Eq. (48) with respect to yi and using boundary condition (22), we obtain
yi
0
H
u3i 1 −
1 (1 + ξ )2
dyi .
(40)
ξ = 0 .
(41)
Integrating with respect to yi and applying boundary conditions (22) and (23) yields h yi + h . H
(42)
Substituting this explicit solution for ξ into Eq. (26) provides the gravity-current velocity h ) 2ˆg h , H
(43)
U 2yi (1 − ), 2 H
(44)
where U denotes the velocity difference between the bottom and top walls at the inlet. With ρi = ρa , Eq. (21) reduces to (45)
By integrating with respect to yi and rearranging, we arrive at ξ (2 + ξ ) C1 = 2 . (1 + ξ )2 ui
(46)
Applying boundary condition (26) for ξ (0) yields U 2 − 2ˆg h . C1 = Ug + 2 c 2016 Royal Meteorological Society
(Ug +
U 2 2 )
− (Ug + 2ˆg h
U 2
−
U 2 H yi )
12 . (49)
Imposing the boundary condition at ξ (H) = 0 in Eq. (49) provides the gravity-current velocity in form h 1 h (50) ) 2ˆg h − (1 − )2 U , H 2 H which is identical to the result obtained by Nasr-Azadani and Meiburg (2015). We remark that Xu (1992) extended Benjamin (1968)’s approach to the same configuration and analyzed flows with and without energy dissipation. For the case of zero head loss, his findings are identical to Eq. (50). We now demonstrate that Benjamin (1968)’s assumption regarding head loss along streamline AO (see Figure 2) being zero is true only for h/H = 1/2. Toward this goal, we first compute the pressure drop along the streamline connecting the bottom wall at the inlet B to the stagnation point O (AO in Figure 2). This can be evaluated using the pressure drop along the top wall (Eq. (36)) and the hydrostatic relations. Since the fluid inside the gravity current is at rest, we apply Bernoulli’s equation along the bottom wall (OO in Figure 2) to obtain pO = pO . Thus, Ug = (1 −
pA − pO = pA − (pA + ρ1 gh) .
(51)
Next, we employ hydrostatic relations (see Eqs (30) and (31)) at the inlet and outlet to find the pressure drop pA − pO : H pA − pO = pB − pB − (ρ1 − ρa ) gh − δρi gξ dyi . (52) 0
For the case of constant density in the ambient studied by Benjamin (1968), the last term in Eq. (52) is equal to zero. By substituting the results obtained in Eqs (36), (42) and (43) into Eq. (52), we obtain the pressure drop along the streamline AO as h h h pA − pO = (ρ1 − ρa )gh − 1 + ρa (2ˆg h) 1 − . 2H H H (53) We can now apply Bernoulli’s equation along the streamline AO to compute head loss via
Ui (yi )
ξ u2i + ξ (1 + ξ )(2 + ξ )ui ui = 0 .
H 2ˆg h U
p O
which is identical to the solution found by Borden and Meiburg (2013a) based on their vorticity model and agrees with the result of Benjamin (1968) when the gravity-current height is half the channel height. For a detailed discussion of the models by Benjamin (1968) and Borden and Meiburg (2013a) and a comparison with Navier–Stokes simulation results, see Borden and Meiburg (2013a). Gravity currents propagating into constant density ambients with uniform shear have been investigated by several authors (see Xu, 1992; Xue et al., 1997; Xue, 2002; Bryan and Rotunno, 2014a,2014b; Nasr-Azadani and Meiburg, 2015). In the steady reference frame, we define the velocity far upstream as ui (yi ) = Ug +
× 1− 1−
We validate our theoretical model and its numerical solution procedure by comparison with two previously well-studied cases, namely a gravity current propagating into constant density fluid at rest and a gravity current moving into a constant density ambient with uniform shear. The classical case of a gravity current entering quiescent constant density fluid has been studied extensively by several authors (e.g. Benjamin, 1968; Shin et al., 2004; Nokes et al., 2008; Borden and Meiburg, 2013a). We recover this case by setting ρi = ρa and ui = Ug , so that Eq. (21) reduces to
Ug = (1 −
ξ (yi ) = −yi + h +
4. Validation and comparison
ξ (yi ) = −
Equation (46) provides a quadratic equation from which ξ is found as ui (yi ) . (48) ξ (yi ) = −1 ±
u2i (yi ) − C1
(47)
1 pA + ρa Ug2 = pO + δAO . 2 The head loss along the streamline AO for this case reads h h −1 . δAO = ρa gˆ h 1 − ( )2 + (ρ1 − ρa )gh H 2H
(54)
(55)
Benjamin (1968) assumes that δAO is always zero for both energyconserving and energy-dissipative flows (see the discussions in sections 2.1 and 2.2 of Benjamin, 1968). Our results obtained from Eq. (55) suggest that the above assumption is valid only for gravity current heights of h/H = 0 and h/H = 0.5. For any other values of gravity-current height h, a non-zero head loss along streamline AO is returned (see figure 6 and the relevant discussion in Borden and Meiburg, 2013a). Q. J. R. Meteorol. Soc. (2016)
M. M. Nasr-Azadani and E. Meiburg
0.4
0.015
0.35
0.01
0.3
0.005
0.25
0
0.2
–0.005
0.15
–0.01
0.1
–0.015
0.05
–0.02
0 0.2
0.3
0.4
0.5
0.6
0.7
0.8
–0.025 0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 3. (a) Gravity-current velocity as a function of the current height. (b) Energy loss (Eq. (40)) as a function of the current height. In both (a) and (b), the inflow density and horizontal velocity profiles are given by Eqs (56) and (57). ‘E0’ indicates the zero energy-loss case, with a current height h∗ = 0.61. ‘Em’ corresponds to the maximum energy-loss case, for which the current height is h∗ = 0.403. To the left of the vertical line, the energy loss is positive, so that the solutions are physically meaningful.
5. Model predictions versus simulation results For given upstream density and velocity profiles, this section discusses the properties of a gravity current as a function of its height and compares model predictions with two-dimensional DNS results. Density outflows from thunderstorms and their interaction with the ambient flow have been suggested as a mechanism for producing long-lived squall lines (cf. Rotunno et al., 1988; Xu and Moncrieff, 1994). Several idealized models have been introduced for such configurations, among them ambient flows with constant shear or a sharp velocity jump (Xu, 1992; Xue et al., 1997; Xue, 2000; Nasr-Azadani and Meiburg, 2015). The present framework allows us to move beyond these highly idealized models and to consider more general nonlinear velocity and density profiles. To illustrate this point, in the following we will focus on ambient velocity and density profiles that decay exponentially across the channel height (for discussion on the influence of stratification on density driven flows, see Moncrieff, 1978; Lilly, 1979; Maxworthy et al., 2002; Ungarish and Huppert, 2002). As a representative example, we choose the profiles ∗
ρi∗ (yi∗ ) = 0.8149e−4yi − 0.0149 , u∗i (yi∗ )
−yi∗
= 0.9027e − Ui∗ (yi∗ )
0.5706 +Ug∗
(56) .
∗
(57)
Hereafter, the symbol refers to a dimensionless quantity. We employ the channel height H, buoyancy velocity gˆ H and density difference (ρ1 − ρa ) as the reference scales and define the dimensionless quantities via x (58) x∗ ≡ , H u u∗ ≡ , (59) gˆ H ρ − ρa , (60) ρ∗ ≡ ρ1 − ρa p , (61) p∗ ≡ ρa gˆ H t t∗ ≡ , (62) H/ gˆ H E˙ E˙ ∗ ≡ . (63) ρa gˆ 3/2 H 5/2 c 2016 Royal Meteorological Society
For the given upstream velocity and density profiles (56) and (57), we evaluate the displacement function ξ ∗ and the gravity-current velocity Ug∗ as functions of the current height h∗ . This is accomplished by solving Eq. (21) numerically based on fourth-order Runge–Kutta integration in conjunction with the iterative procedure discussed in section 2. Figure 3(a) depicts the gravity-current velocity Ug∗ as a function of the current height h∗ . We observe a maximum at h∗ ≈ 0.4. The location of this maximum and its value depend strongly on the ambient shear and stratification. To assess whether or not the obtained solutions are physically feasible, Figure 3(b) shows the corresponding energy loss δ E˙ ∗ (see Eq. (40)). The zero energy-loss case is indicated by the vertical dotted line. To the left of this line, i.e. for h∗ ≤ 0.61, the energy loss is positive, which indicates that the solutions are physically meaningful. For h∗ > 0.61, on the other hand, negative energy-loss values suggest that the corresponding flows cannot be realized without an external energy supply. As discussed in section 2, the present analysis applies to supercritical gravity currents, which give rise to steady-state conditions in the moving reference frame. Therefore, the results obtained should always satisfy Fr > 1, where the Froude number is defined according to Eq. (27). As discussed before, for a linearly stratified ambient without any shear, gravity currents with velocity Ug∗ > 1/π are supercritical. For a nonlinear density distribution in the ambient, however, the coefficient α in Eq. (27) related to the first-mode long-wave speed (cf. White and Helfrich, 2008) needs to be computed by solving a linear eigenvalue problem for ρ(y). Furthermore, as discussed by Baines (1998), ambient shear can also alter the characteristic wave speed (and therefore the Froude number). Hence, the condition for supercriticality has to be evaluated separately for every combination of ambient density and velocity profiles. We now proceed to compare the model predictions with DNS simulation results. Towards this end, we initialize the flow field with a current that has a height corresponding to zero energy loss, as indicated by E0 in Figure 3. This case corresponds to h∗ = 0.61 and Ug∗ = 0.3. The initial conditions in the DNS simulations are as follows. The density field takes the inlet density profile up to the front location. Downstream of the front, we set the density in the ambient by the predictions made by our model. As for the velocity field, we initially set the vertical component to zero everywhere. The horizontal component is set to the prescribed inlet velocity upstream of the gravity current front and to the Q. J. R. Meteorol. Soc. (2016)
Gravity Currents in Sheared Stratified Ambients
Figure 4. Density fields shown at different times for case E0 with Re∗ = 2500. The initial density and horizontal velocity fields are set to the values predicted by our model. The horizontal dashed line indicates the height predicted by the model for zero energy loss (h∗ = 0.61). After the initial transient phase, the gravity current reaches a quasi-steady state with a current height close to the predicted value.
Figure 5. Steady-state streamlines along with the density field for a DNS simulation with Re∗ = 2500 (see Figure 3). The quasi-steady current height is close to the value 0.61 predicted by the model for a flow with zero energy loss (case E0).
values predicted by the model downstream thereof. Inside the gravity current, the horizontal and vertical velocities are initially set to zero. We remark that the initial shape of the the interface in the vicinity of the front location may influence the transient stage, but it does not affect the quasi-steady results. We have verified this by choosing an error function instead of a sharp jump at the interface and obtained identical results (see also the discussion on the re-initialization procedure in Nasr-Azadani and Meiburg, 2015). The simulation is carried out in the reference frame moving with the current velocity predicted by the model. Hence, the c 2016 Royal Meteorological Society
agreement between simulation results and model predictions can be assessed by the degree to which the current front remains in place throughout the simulation. We allow the flow to develop in time until a nearly steady state is reached. The simulation is performed by our in-house code TURBINS (Nasr-Azadani and Meiburg, 2011). TURBINS is a finite-difference code, which employs TVD-RK3 temporal integration along with a fractionalstep projection method to solve the Navier–Stokes equations in the Boussinesq approximation. A detailed description of TURBINS, along with validation results and comparisons against experiments, is presented in Nasr-Azadani and Meiburg (2011) Q. J. R. Meteorol. Soc. (2016)
M. M. Nasr-Azadani and E. Meiburg The density field ρ ∗ (x∗ , t ∗ ) is described by the convection–diffusion equation ,
2.2
∂ρ ∗ 1 + u∗ · ∇ρ ∗ = ∗ ∗ ∇ 2 ρ ∗ . ∂t ∗ Re Sc
2
In the simulation, we employ Sc∗ = 6 and we refer the reader to H¨artel et al. (2000) for a discussion of the influence of the Schmidt number on the dynamics of gravity currents. In the simulations, we separately keep track of the density fields associated with the ambient (ρ0∗ (x∗ , t ∗ )) and with the gravity current (ρ1∗ (x∗ , t ∗ )). The total density field is then obtained by superimposing the two: ρ ∗ x∗ , t ∗ = ρ0∗ x∗ , t ∗ + ρ1∗ x∗ , t ∗ . (68)
1.8
1.6
1.4
The main reason for distinguishing these two density fields is to compute the current height h¯ ∗ by means of the depth-integrated density field ρ1∗ associated with the gravity current:
1.2
1
(67)
0
10
20
30
40
50
60
70
80
h¯ ∗ (x∗ , t ∗ ) =
0
1
ρ1∗ (x∗ , y∗ , t ∗ ) dy∗ .
(69)
Figure 6. Temporal evolution of the front location xf∗ recorded from the DNS simulation E0 (see Figure 5) for various Reynolds numbers. The front velocity u∗f is equal to 0.005, 0.0055 and 0.0066 for Re∗ values equal to 2500, 10 000 and 20 000, respectively. In all cases, it causes only a small relative error of approximately 1–2% compared with the analytical value of Ug∗ = 0.30.
The computational domain has dimensions L∗x × L∗y = 4.5 × 1, with in- and outflow boundaries in the horizontal direction. We choose uniform grid spacings in the x∗ - and y∗ -directions, with x∗ = 0.0128 and y∗ = 0.005. For the density field, we specify no-flux conditions at the top and bottom walls. At those walls, 2 ∗ ∗2 and Nasr-Azadani et al. (2013), so that a brief summary will we furthermore impose ∂ u /∂y = 0, so that the slope of the velocity profile is free to adjust itself. This boundary condition was suffice here. We employ the two-dimensional incompressible Navier– chosen because (i) it ensures that the influence of the boundaries on the flow in the interior of the domain is very weak and (ii) it Stokes equations enforces a vanishing vorticity flux across the horizontal walls. At ∇ · u∗ = 0 , (64) the downstream boundary, we convect all flow variables q∗ across the boundary via the non-reflective outflow condition 1 2 ∗ ∂u∗ ∗ ∗ ∗ ∗ g + u · ∇u = −∇p + ∇ u + ρ e , (65) ∂t ∗ Re∗ ∂q∗ ∂q∗ + U¯ ∗ ∗ = 0 , (70) g ∗ to describe the fluid motion. Here, e and Re∗ , respectively, denote ∂t ∂x the unit vector acting in the direction of gravity and the Reynolds number, where U¯ ∗ represents the maximum u∗ velocity value in the domain. H gˆ H Figure 4 shows the transient evolution of the current initialized Re∗ = . (66) ν with the density and velocity fields predicted by our model.
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 –0.2
0
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Figure 7. Vertical distribution of (a) horizontal velocity and (b) density plotted at x∗ = 4 obtained from simulation E0. Very good agreement is observed between the DNS findings (dashed lines) and theoretical predictions (solid lines). For the sake of comparison, the prescribed inlet density and velocity profiles are also shown by dash–dotted lines.
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Q. J. R. Meteorol. Soc. (2016)
Gravity Currents in Sheared Stratified Ambients
1
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Figure 8. (a) Vertical distribution of streamline displacement function ξ ∗ and outlet streamline location yo∗ computed for case E0. (b) Outlet to inlet velocity and vorticity ratio plotted for problem E0. With the presence of background stratification and the resulting baroclinic vorticity, the outlet vorticity changes sign at ξ ∗ = 0.36 (yi∗ = 0.57), which causes a maximum in the velocity at the outlet region. The RK4 numerical method is employed to solve the ODE in Eq. (21). Bullets and diamonds are computed from the DNS results.
(a) Ambient density
(b) Gravity current density
(c) Total density
Figure 9. Density fields shown at t ∗ = 60 for the case with Re∗ = 10 000. The horizontal dashed line indicates the height predicted for zero energy loss (h∗ = 0.61). As a result of unsteadiness, the average gravity-current height falls below this value, while the gravity-current velocity remains close to the value predicted for zero energy loss.
Upon release, the flow adjusts itself quickly to form a gravity current front, while exhibiting interfacial instabilities in the region separating the gravity current and the ambient. Depending on the value of the Reynolds number, the interface separating the gravity current and the ambient fluid can exhibit absolutely or convectively unstable behaviour (Huerre and Monkewitz, 1990). Since the present investigation focuses on steady gravity currents, we set the Reynolds number to 2500, which is close to the highest value for which all interfacial perturbations are convected out of the domain, so that a nearly steady state develops. The case of higher Reynolds numbers and unsteady interfaces subject to Kelvin–Helmholtz instabilities will be discussed below. Figure 5 visualizes the steady-state density field along with the streamlines at time t ∗ = 60. Downstream of the frontal c 2016 Royal Meteorological Society
region, the gravity-current height approaches the analytical value corresponding to zero energy loss (shown by the dashed line) very closely. Figure 6 shows the front location xf∗ as a function of time in the moving reference frame. After an initial transient period, the flow reaches quasi-steady conditions characterized by a small, approximately constant velocity of u∗f = dxf∗ /dt ∗ ≈ 0.005 (solid line in Figure 6). This indicates that the relative difference between the model prediction for zero energy loss (Ug∗ = 0.3) and the front velocity recorded in the viscous Navier–Stokes simulation (Ug∗ = 0.3 − u∗f ) is of the order of 1.6%, which we take as good agreement between model prediction and simulation result. Due to the existence of viscous effects in the DNS simulation, a weak internal recirculation inside the gravity-current body Q. J. R. Meteorol. Soc. (2016)
M. M. Nasr-Azadani and E. Meiburg 1 0.8 0.6 0.4 0.2 0
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Figure 10. Solid line: depth-integrated height h¯ ∗ of the steady gravity current for Re∗ = 2500 (see Eq. (69)). Dotted line: temporally averaged height h¯ ∗ of the unsteady gravity current for Re∗ = 10 000 (see Eq. (71)). The height of the steady current agrees closely with the value 0.61 predicted for zero energy loss (dashed horizontal line), while the temporally averaged height of the unsteady current is close to 0.403, which is the value predicted for maximum energy loss (horizontal dash–dotted line). The error bars correspond to the maximum and minimum gravity-current height within the temporal averaging interval.
forms, as shown in Figure 5 and also discussed in Nasr-Azadani and Meiburg (2015). Xu and Moncrieff (1994) investigate this recirculation theoretically and demonstrate that the influence of the internal circulation on the frontal region and the propagation velocity of the gravity current is small. We remark that the current theoretical framework could be extended to include this effect by adding a constant shear inside the gravity current. Figure 7 compares the quasi-steady density and velocity profiles near the downstream boundary with the model predictions. For completeness, the prescribed inflow velocity and density profiles are shown by dash–dotted lines. Away from the interface, the simulation results duplicate the model predictions closely. Near the interface, the discontinuities in u∗ and ρ ∗ are smeared out by the presence of diffusion in the simulation. Interestingly, the downstream velocity exhibits a local maximum value at yo∗ = 0.93, which corresponds to ξ ∗ = 0.36, cf. Figure 8(a). This height corresponds to the point where the outflow to inflow vorticity ratio ωo∗ /ωi∗ = 1 − ξ ∗ ρ ∗ /{u∗ u∗ } reaches a zero value (see the dashed line in Figure 8(b)). This change in sign of the outflow vorticity ratio and the corresponding outflow velocity maximum is achievable only due to the existence of background stratification.The bullets and diamonds shown in Figure 8 are from the DNS results. Despite slight differences at the interfacial region, due mainly to viscous and unsteady effects, our theoretical predictions follow the DNS results very closely. 5.1.
Higher Reynolds numbers and unsteady interfaces
In natural settings, gravity currents give rise to much larger Reynolds numbers than the ones considered so far. Under such conditions, they typically exhibit unsteady dynamics and reduced heights, as reported by Benjamin (1968), Xu (1992), Bryan and Rotunno (2014a) and Nasr-Azadani and Meiburg (2015) for different situations. Based on two-dimensional DNS simulations, Nasr-Azadani and Meiburg (2015) demonstrate that, for gravity currents in sheared environments, the presence of Kelvin–Helmholtz instabilities results in gravity-current heights corresponding to maximum energy dissipation. This hypothesis has been modelled (Xu, 1992) and compared with two- and threedimensional simulations by Bryan and Rotunno (2014a). Xu (1992) suggests an empirical correlation for the head loss across the channel height and studies gravity currents with unstable interfaces. Bryan and Rotunno (2014a) study three-dimensional gravity currents with maximum energy dissipation based on Xu (1992)’s model. While the front velocity obtained from these modelling efforts exhibits good agreement with simulation results, the reduced gravity-current height agrees less well with the theoretical predictions. In the following, we discuss two DNS simulations with Re∗ values of 10 000 and 20 000, respectively. To reduce the effect of the outflow boundary on the flow dynamics and to resolve the sharp interface, we choose a bigger computational box with L∗x × L∗y = 9 × 1 and a finer grid spacing of x∗ = 0.009 and y∗ = 0.004. The front location as a function of time for each case is depicted in Figure 6, in the reference frame moving with the velocity Ug∗ = 0.30 of the zero energy-loss case. Figure 9 depicts the density fields associated with the gravity current and the ambient for the case Re∗ = 10 000. Near the front, the c 2016 Royal Meteorological Society
gravity-current height appears to approach the zero energy-loss value. Further downstream, however, strong Kelvin–Helmholtz instabilities emerge and the current height (Eq. (69)) decreases. Figure 10 verifies that, for the stable interface case of Re∗ = 2500, the gravity-current height approaches the predicted zero energyloss value (dashed line in Figure 10). To compare this case with the higher Reynolds-number scenario, we compute the temporally averaged current height h¯ ∗ (x∗ ) : h¯ ∗ (x∗ ) =
1 te∗ − ts∗
te∗
ts∗
h¯ ∗ (x∗ , τ ∗ ) dτ ∗ .
(71)
Here, ts∗ and te∗ , indicate the temporal interval over which the current height is averaged. We set their values to 65 and 80, respectively, during which the current has reached statistically steady conditions. Near the front, the temporally averaged current height (dots in Figure 10) approaches the zero energy-loss value. Further downstream, however, it drops to a significantly lower value. Xu (1992), Bryan and Rotunno (2014a) and Nasr-Azadani and Meiburg (2015) suggest that, for gravity currents interacting with ambient shear, this reduced value should correspond to the height at which energy dissipation reaches a maximum. Our energy analysis places this maximum at a value of h∗ = 0.403 (Em in Figure 3), which is shown by the dash–dotted line in Figure 10. The DNS result is seen to lie close to this value, which supports the hypothesis that the current height approaches the value associated with maximum energy loss. 6. Summary and concluding remarks Within the present article, we present the development of a vorticity-based, inviscid model for steady Boussinesq gravity currents interacting with arbitrary background velocity shear and density stratification. This model extends the approach first introduced by Borden and Meiburg (2013a) for gravity currents propagating into an unstratified ambient fluid at rest. Given the gravity-current height and upstream velocity and density profiles, a second-order nonlinear ODE is obtained that can be solved for the streamline displacement and the gravity-current velocity, along with the downstream density and velocity profiles. The model does not require any empirical closure assumptions. The energy and head loss of the flow can be analyzed a posteriori, in order to assess the feasibility of the solutions. For representative nonlinear shear and density distributions, the model is compared with two-dimensional Navier–Stokes simulation results. For moderate Reynolds numbers and a stable interface separating the ambient region from the gravity current, we observe the evolution of a quasi-steady current, the properties of which agree closely with the model predictions for energyconserving flow. Consistent with several previous observations (Benjamin, 1968; Xu, 1992; Borden and Meiburg, 2013a; Bryan and Rotunno, 2014a; Nasr-Azadani and Meiburg, 2015), we find that for high Reynolds numbers the interface is strongly influenced by unsteady Kelvin–Helmholtz instabilities. While the propagation velocity of the gravity current remains largely unchanged under such conditions, the temporally averaged gravity-current height adjusts itself to a lower value that agrees Q. J. R. Meteorol. Soc. (2016)
Gravity Currents in Sheared Stratified Ambients closely with the one predicted for maximum energy loss (Xu, 1992; Bryan and Rotunno, 2014a; Nasr-Azadani and Meiburg, 2015). Acknowledgement We gratefully acknowledge support through NSF grant CBET1335148. Appendix A In section 2, we developed a vorticity-based model for continuous ambient density and velocity profiles. Here, we extend this framework to scenarios in which the ambient density/velocity profiles exhibit jumps in their values and/or derivatives. Toward this end, we assume that the ambient consists of two separate layers (see Figure A1), with continuous velocity and density distributions within each layer. The discontinuity in the density and velocity profiles gives rise to a vortex sheet separating these two layers (DD in Figure A1). Quantities in the lower and upper layers are denoted by subscripts 1 and 2, respectively. Let us define the streamline displacement functions ξ1 and ξ2 for the lower and upper layers, respectively. In each ambient layer, the second-order ODE governing the displacement function (see Eq. (21)) is governed by ξ1 u2i1 + ξ1 (1 + ξ1 )(2 + ξ1 )ui1 ui1 − ξ1 (1 + ξ1 )3
gˆ ρi1 = 0, (ρ1 − ρa )
(A1)
gˆ ρi2 = 0. (ρ1 − ρa )
2
(A2)
1
ρi1 (Y1 ) − ρi2 (Y1 ) ξ1 (Y1 ) . − gˆ ρ1 − ρa
(A6)
We now assume that ξ1 and ξ1 are already found using the solution of the ODE (A1) with appropriate boundary conditions (A3) and (A4). Given that the inlet velocity and density profiles are also known quantities, we proceed to obtain ξ2 (Y1 ) by solving the quadratic form of Eq. (A6): ξ2 (Y1 ) = −1 ± 1+
u2i2 (Y1 )
+
ui1 ui2
1 2
1 {1+ξ1 (Y1 )}2
, −1
(A7)
with denoting = −2ˆg
ξ2 u2i2 + ξ2 (1 + ξ2 )(2 + ξ2 )ui2 ui2 − ξ2 (1 + ξ2 )3
We note that the rate at which a vortex sheet transports circulation is given by the vortex sheet’s principal velocity multiplied by the vortex sheet strength (Saffman, 1992). The last term in Eq. (A5) denotes the baroclinic vorticity generation across the interface DD . Next, we employ Eq. (19) to relate the oulet velocity to the inlet velocity in each layer and simplify Eq. (A5) to the following form: 1 2 ui2 (Y1 ) − u2i1 (Y1 ) 2 u2i2 (Y1 ) u2i1 (Y1 ) 1 − = 2 2 1 + ξ (Y1 ) 2 1 + ξ (Y1 )
ρi1 (Y1 ) − ρi2 (Y1 ) ξ1 (Y1 ) . ρ1 − ρa
(A8)
Geometrical considerations (see Figure A1) provide us with the two additional boundary conditions:
To solve the above set of equations, each ODE requires two boundary conditions. For the layer adjacent to the gravity current (layer 1), the boundary conditions presented in Eqs (22) and (26) can be employed as follows:
ξ2 (yi = Y1 ) = ξ1 (yi = Y1 ) ,
ξ1 (yi = 0) = h , Ui1 (0) + Ug − 1. ξ1 (yi = 0) =
i1 (0) 2ˆg h · ρ1ρ−ρ 1 −ρa
(A4)
Equations (A3), (A4), (A7), (A9) and (A10) provide five boundary conditions, which can be employed to solve the two second-order ODEs given in Eqs (A1) and (A2) along with the gravity-current velocity Ug . For the general case, the solution procedure is as follows.
To obtain the required boundary conditions for the upper layer (layer 2), we enforce the conservation of vorticity within a narrow control volume containing the interface DD (Figure A1), in the following form: 1 ui2 (Y1 ) − ui1 (Y1 ) × ui2 (Y1 ) + ui1 (Y1 ) 2 1 = u2o (Y1o ) − uo1 (Y1o ) × uo2 (Y1o ) + u1o (Y1o ) 2 ρi1 (Y1 ) − ρi2 (Y1 ) × (Y1o − Y1 ) . (A5) − gˆ ρ1 − ρa
(1) Guess the value of the gravity-current velocity Ug . (2) Compute ξ1 (0) from Eq. (A4). (3) Solve Eq. (A1) to compute ξ1 , based on the two initial conditions ξ1 (0) and ξ1 (0) (Eqs (A3) and (A4)). (4) Update the boundary condition required for the upper layer 2 via Eq. (A7) and the solution in the lower layer 1. (5) Solve Eq. (A2) to compute ξ2 , based on the two initial conditions ξ2 (Y1 ) and ξ2 (Y1 ) (Eqs (A7) and (A9)). (6) Check whether or not the numerical value for ξ2 (H) satisfies the boundary condition (A10). (7) Update Ug and repeat the loop until condition 6 is satisfied.
(A3)
ξ2 (yi = H) = 0 .
(A9) (A10)
Figure A1. Configuration of a gravity current (shown in grey) propagating into a sheared and stably stratified two-layer ambient flow. Within each layer, the velocity and density profiles are continuous functions of the inlet height yi .
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