Maximum Penetration Height and Intrusion Speed of Weak Symmetric Plane Fountains in Linearly Stratiﬁed Fluids

: The ﬂow behavior of weak symmetric plane fountains in linearly stratiﬁed ﬂuids is studied numerically with three-dimensional simulations over a range of the Froude ( Fr ), Reynolds ( Re )


Introduction
Fountains are widely encountered in many natural settings and practical applications.Their behavior is also of fundamental significance as our understanding of free shear flows under negatively buoyant force still needs improvement.These make them a topic that has drawn a remarkable research interest.Hunt and Burridge [1] conducted a very comprehensive review on the studies on the behavior of different types of fountains over wide ranges of governing parameters and under different conditions.
A fountain is a flow caused by a fluid injected vertically upward into a large body of fluid which has a smaller density.The buoyancy experienced by the upward fluid opposes its upward velocity resulting in a gradual reduction in velocity up to zero.The height where the upward velocity attains zero is denoted as the maximum fountain penetration height (abbreviated as 'MFPH' hereafter).After the upward fluid attains the MFPH, it reverses the moving direction and descends and falls on the bottom floor to form an intrusion that moves outward along the bottom floor.
The ambient fluid in which a fountain penetrates can be homogeneous or stratified.When the ambient fluid is stratified, the fluids at different heights have different densities.A stable stratification with a constant stratification number is the physical situation in which the fluid densities decrease from the bottom to the top at a constant density gradient.Such a fluid is called a linearly stratified fluid.A fountain is generally characterized by the Reynolds number, Re, Froude number, Fr, and density stratification parameter, S p , if the ambient is a linearly stratified fluid (abbreviated as 'LSF' hereafter).These three governing parameters are defined below [2], Fr in the second expression of Equation ( 2) is implemented when the Oberbeck-Boussinesq approximation is valid, which is what is assumed in the present study.The temperature stratification parameter, S, can also be represented using this approximation, as follows [2], The dimensionless form of S is widely used to generalize the results and defined below [2], Past studies on fountains have focused on round ones in homogeneous fluids (i.e., S p = 0), with the dimensionless MFPH (z m ) as the main variable to characterize and quantify the fountain behavior.A round fountain can be 'very weak' when Fr 1, 'weak' when 1 Fr 3, or 'forced' when Fr 3 [3,4].Forced round fountains are found to be significantly different from weak fountains.The readers are referred to the review by Hunt and Burridge [1] for the details.
Similar to round fountains, a PF in a large container of homogeneous fluid may be either 'very weak' when Fr 1, 'weak' when Fr = O(1), or 'forced' when Fr 1, as classified by Hunt and Coffey [11].
The studies on PFs have dominantly focused on fountains in homogeneous fluids, although some studies also focused on those in stratified fluids, such as [5,8,[12][13][14][15].Research on PFs in stratified fluids has been scarce and mainly focused on those in turbulent regimes, with z m as the main parameter.In the past several years, we carried out numerical studies on weak and transitional PFs in LSFs to examine their effects on the transition of a PF from symmetric behavior to asymmetric behavior, with z m as the major parameter characterizing the PF behavior [10,[16][17][18].Nevertheless, the characterization of weak PFs in stratified fluids with small Fr and Re values, in which only the symmetric behavior is present, is currently rarely understood.Furthermore, the intrusion is an integral part of a weak symmetric PF (abbreviated as 'SPF' hereafter), and its behavior is important due to its significant effects on z m .Nonetheless, intrusion in weak SPF, particularly in stratified ambient fluids, is rarely investigated.The aforementioned points motivate us to investigate the characterization of penetration and intrusion of SPFs in LSFs.
In this work, numerical simulations were executed for weak SPFs in a large body of LSF over a range of Fr and s as a fixed small Re to ensure the fountains were laminar and to study their effects on the MFPH and the intrusion behavior of weak SPFs in LSFs.

Methodology
The physical and numerical model is assumed to be a rectangular domain with the dimensions H × B × L, in which an initially quiescent fluid at a constant stratification parameter value is fully filled.At the bottom center of the domain, as the source for the PF, a slot with a half-width of X 0 exists.The remainder of the bottom and top surfaces (in the X-Y plane, at Y = H) are rigid, non-slip and adiabatic.The periodic boundary condition is applied to the two vertical sidewalls (in the X-Z plane, at Y = ±B/2), and the two vertical surfaces in the Y-Z plane, at X = ±L/2 (front and back boundaries), are considered to be outflows.The origin of the Cartesian coordinate systems is at the bottom center, with gravity in the negative Z-direction.At time t = 0, to initiate the PF flow, a fluid at the temperature T 0 , which is smaller than T a,0 , with a uniform velocity W 0 is injected upward from the slot into the domain, which is subsequently continued during the whole time period of a particular numerical simulation run.

Development of Temperature Field
The development of the temperature field in the domain is shown by the evolution of the transient temperature contours of the PF at Fr = 3.25 and s = 0.1 on three specific planes, as illustrated in Figure 1.It can be observed that for all the instants illustrated, the X = 0 surface in the Y-Z plane is the symmetry surface (as shown in the first column of Figure 1).From the second and third columns, it can be seen that along the Y direction, the temperature gradient is zero.This observation is unlike the asymmetric behavior reported for PFs in stratified fluids at larger Fr and Re [10].This finding implies that for the specific fountain presented, the flow and temperature pattern remains symmetric irrespective of the instant selected in contrast to an asymmetric fountain [10,16,17].

Effect of Fr
Figure 2 demonstrates the influence of Fr on the SPFs at the fully developed stage (abbreviated as 'FDS' hereafter).The symmetry observed for different instances in Figure 1 can be seen for different Fr cases.By increasing Fr, the momentum flux of the injected fluid becomes larger, which leads to increased z m .A typical feature of a weak fountain is having indistinguishable upflow and the downflow that makes little entrainment of the ambient fluid into the core region of the fountain fluid [2,6].The first and second columns of Figure 2 also depict the variation of intrusion thickness.The intrusion thickness increases considerably compared to the fountain height, especially for a small Fr.This behavior has a major impact on z m that will be discussed subsequently.

Effect of s
To investigate the influence of s, which is the dimensionless stratification number defined by Equation (5), Figure 3 illustrates the temperature contours of five SPFs at the FDS.Similar behavior to that in Figures 1 and 2 can be identified, as fountains stay symmetric and the mixing between the fountain and fluid is minimal.The intrusion again increases significantly compared to the fountain height, especially for a large s.z m reduces with increasing s because of the stabilizing ability of the stratification as discussed in [10,16,17], indicating that s has a notable influence on the MFPH along with the intrusion height.

Quantitative Analysis 4.1. MFPH (z m )
A sample of the development of the numerically obtained z m and the corresponding velocity v m , which is made dimensionless by W 0 , are shown in Figure 4 for the SPF at Fr = 2 and s = 0.1.From Figure 4a, it can be seen that there is a continuous rise of fountain height until about τ = 10, when the fountain reaches its initial MFPH, z m,i .At this time (τ m,i ), as expected, v m decreases to zero for the first time since the commencement of the fountain as presented in Figure 4b.Then, after a slight fall, z m rises and this growth continues at very small rates, which vary with time, and the flow is at the FDS, as can be seen in Figure 4b.This is distinctly different from the behavior of an asymmetric PF that when the fountain height reaches z m,i , after a transition period, z m oscillates around an almost fixed value (z m,a ) at its FDS as shown in [10,16].The observed continuous increase of the MFPH in an SPF is largely caused by the intrusion which continuously diminishes with time at the FDS, leading to reduced buoyant force experienced by the fountain fluid.It is important to quantify the MFPH at the FDS.Therefore, it was decided to find the time-averaged value of z m over a specific time period (the 'averaging period'), denoted as z m,a , along with the MFPH at the commencement of the time averaging period, z m,s , as depicted in Figure 4a.The time-averaged velocity v m,a during the averaging period is apparently the appropriate parameter to quantify the extent of the variation of z m at the FDS.In this study, for consistency, the time instant for z m,s is at τ = 100, while the averaging period for both z m,a and v m,a is over 100 ≤ τ ≤ 900 for all fountains considered. 1, where z m,i is the initial MFPH and τ m,i is the time instant when z m = z m,i , z m,s is the MFPH at the commencement time of the period for the time averaging of z m at the FDS which gives the time-averaged MFPH z m,a .All heights, times and velocity are made dimensionless by X 0 , X 0 /W 0 and W 0 , respectively.The averaging period for both z m,a and v m,a is 100 ≤ τ ≤ 900.
The numerical obtained z m and v m for five SPFs with five Fr values at s = 0.1 and five SPFs with five s values at Fr = 2 are shown in Figure 5.It is clearly seen that both Fr and s have significant effects on z m and v m .As expected, z m increases when Fr becomes larger, but it decreases when s increases, as the corresponding negative buoyancy increases too, which leads consequently to smaller τ m,i , z m,s , z m,a and v m,a , as will be shown subsequently.
The effects of Fr and s on z m,i are illustrated by the numerical results shown in Figure 6 for all SPFs considered in the present study.As shown in Figure 6a, for each s value, z m,i increases when Fr becomes larger, which can be quantified by an approximately quadratic correlation, owing to the increased momentum flux of the fountain.It is also observed that for each Fr, the increase of s results in reduced z m,i , as also shown in Figure 6b, resulting from a stronger negative buoyancy.
For a weak SPF in an LSF, Lin and Armfield [2] asserted that momentum and buoyancy fluxes, kinematic viscosity of fluid, and ambient stratification together form a complete parametrization of the MFPH, and they derived the scaling relation for z m through a dimensional analysis, as shown below, where the constants b and c can be obtained empirically with experimental or numerical results.They confirmed this scaling relation for weak SPFs with a series of numerical simulations with varied Fr, Re and s values.For the weak SPFs considered in the present study, with Re fixed at Re = 100, it is impossible to obtain the value of c, but apparently, the overall effects of Fr and s on z m,i can be represented by Fr a s b , where a is another constant which can be determined empirically.
With the numerical results obtained for all fountains considered here, a multi-variable regression analysis gives a = 0.79 and b = −0.178for z m,i , and the following correlation is obtained to quantify the overall effects of Fr and s on z m,i , z m,i = 0.22(Fr 0.79 s −0.178 ) 2 + 0.46Fr 0.79 s −0.178 + 1.106 = 0.22Fr 1.58 s −0.356 + 0.46Fr 0.79 s −0.178 + 1.106, (7) with the regression constant of R 2 = 0.9947.This indicates that Fr 0.79 s −0.178 quantifies the overall effects of Fr and s on z m,i very well, as also shown in Figure 6c.Nevertheless, it was noted from the numerical simulations that among all fountain cases considered, there are four cases at high Fr and s values that become asymmetric after z m attains z m,i , although only slightly.These cases are those at Fr = 4.5 and Fr = 5 with s = 0.4 and at Fr = 4.85 and Fr = 5 with s = 0.5.Hence, the results of these four cases should be excluded in the multi-variable regression analysis to obtain the correlation to quantify the combined effects of Fr and s.With the exclusion of the results of these four cases, it is found that a and b have very small changes from those without the exclusions, i.e., a changes from 0.79 to 0.768, and b changes from −0.178 to −0.188.The following corresponding correlation is then obtained with Fr 0.768 s −0.188 , z m,i = 0.214(Fr 0.768 s −0.188 ) 2 + 0.535Fr 0.768 s −0.188 + 0.99 = 0.214Fr 1.536 s −0.376 + 0.535Fr 0.768 s −0.188 + 0.99, with R 2 = 0.9963, indicating that Fr 0.768 s −0.188 provides a slightly better representation of the overall effects of both Fr and s on z m,i , as shown in Figure 6d.A local Froude number at height z can be introduced, which is defined as It can be found that It is appropriate to assume that the effect of s is incorporated in Fr(z).Hence, it is expected that z m,i depends on Fr(z m,i ) only and the scaling relation z m,i ∼ Fr developed for weak SPFs in homogeneous ambient fluids by Lin and Armfield [6] and others should be applicable here as well if Fr is replaced by Fr(z m,i ), as validated by the numerical results shown in Figure 6e,f.From the results for all fountains considered, the following linear correlation is obtained with a linear regression analysis, with R 2 = 0.9521, indicating that the scaling relation z m,i ∼ Fr(z m,i ) is applicable well for the majority of the fountains considered, as shown in Figure 6e.Similarly, the results for the four cases mentioned above should be excluded, which changes the correlation to be with R 2 = 0.9793.It is clearly seen that the scaling relation z m,i ∼ Fr(z m,i ) is applicable very well for all fountains with the exclusion of the four fountains, as indicated in Figure 6f.
As z m,s and z m,a are essentially the parameter as z m,i to represent the MFPH and differ from z m,i only in that they quantify the MFPH at different times, it is expected that the characteristics of z m,s and z m,a should be the same as that of z m,i and all conclusions obtained above for z m,i are applicable to z m,i too.However, there are quantitative differences in the correlations for z m,s and z m,a , as shown in Figures 7 and 8.With the numerical results obtained for all fountains considered, the correlations are obtained for z m,s and z m,a , both without the exclusion of any fountains and with the exclusion of the four fountains mentioned above, and these are listed in Table 1.As shown in Figures 7 and 8, all these correlations indicate that, similar to that for z m,i , the respective Fr a s b obtained quantifies the overall effects of Fr and s on z m,s and z m,a very well, with those with the exclusion of the four fountains providing a slightly better representation of the overall effects of both Fr and s.
Table 1.Numerical obtained empirical correlations for z m,s and z m,a with and without the exclusions.

Correlation
Exclusion?R 2 z m,s = 0.316Fr 1.49 s −0.342 + 0.392Fr 0.745 s −0.171 + 1.434No 0.9938 z m,s = 0.411Fr 1.46 s −0.356 + 0.022Fr 0.73 s −0.178 + 1.754 Yes 0.9956 z m,s = 2.855Fr(z m,s ) − 0.0873 No 0.9578 z m,s = 2.696Fr(z m,s ) + 0.113 Yes 0.9684 z m,a = 0.428Fr 1.318 s −0.328 + 0.514Fr 0.659 s −0.164 + 1.682No 0.9958 z m,a = 0.523Fr 1.296 s −0.338 + 0.179Fr 0.648 s −0.169 + 1.949 Yes 0.9967 z m,a = 2.940Fr(z m,a ) + 0.585 No 0.9762 z m,a = 2.814Fr(z m,a ) + 0.738 Yes 0.9830 A similar analysis can be made on τ m,i , which is another key parameter.The effects of Fr and s on τ m,i are indicated by the numerical results shown in Figure 9 for all fountains considered.Similar to that for z m,i , as shown in Figure 9a, for each s value, τ m,i increases with increasing Fr, which can also be quantified by an approximately quadratic correlation because of the larger momentum flux, and for each Fr value, the increase of s results in reduced τ m,i , which can be seen from Figure 9b as well because of the increased negative buoyancy.
Similar to that for z m,i , the overall effects of Fr and s on τ m,i can be quantified by Fr a s b .However, apparently, it is expected that the values of a and b obtained from the numerical results should be significantly different from those for z m,i .These are confirmed by the results presented in Figure 9c-f.A multi-variable regression analysis of the results for τ m,i from all fountains considered shows that Fr 1.273 s −0.229 quantifies the overall effects of Fr and s on τ m,i well, with the following correlation obtained when all fountains are included, τ m,i = 0.209(Fr 1.273 s −0.229 ) 2 + 1.879Fr with R 2 = 0.9752, as shown in Figure 9c.Similarly, the results of the four fountains mentioned above should be excluded in the regression analysis, and it is found that the obtained Fr 1.225 s −0.252 provides a slightly better representation of the overall effects of Fr and s on τ m,i , with the following correlation obtained when the four fountains are excluded, τ m,i = 0.106(Fr 1.225 s −0.252 ) 2 + 2.897Fr 1.225 s −0.252 + 0.855 = 0.106Fr 2.45 s −0.504 + 2.897Fr 1.225 s −0.252 + 0.855, with R 2 = 0.9827, as shown in Figure 9d.For weak symmetric fountains in homogeneous ambient fluids, Lin and Armfield [6,19] show that τ m,i ∼ Fr 2 .With the effect of s incorporated in Fr(z), similar to that for z m,i , it is expected that τ m,i depends on Fr(z m,i ) only and the scaling relation τ m,i ∼ Fr 2 developed by Lin and Armfield [6,19] should be applicable here as well if Fr is replaced by Fr(z m,i ).This is validated by the results shown in Figure 9e,f.From the results for all fountains considered, the following correlation is obtained with a regression analysis,  (c) z m,a plotted against Fr 0.659 s −0.164 over 0.1 ≤ s ≤ 0.5 and 1 ≤ Fr ≤ 5, without exclusions, (d) z m,a plotted against Fr 0.648 s −0.169 over 0.1 ≤ s ≤ 0.5 and 1 ≤ Fr ≤ 5, with exclusions, (e) z m,a plotted against Fr(z m,a ) over 0.1 ≤ s ≤ 0.5 and 1 ≤ Fr ≤ 5, without exclusions, (f) z m,a plotted against Fr(z m,a ) over 0.1 ≤ s ≤ 0.5 and 1 ≤ Fr ≤ 5, with exclusions, respectively.τ m,i = 10.246Fr(zm,i ) 2 − 17.263Fr(z m,i ) + 13.198, (15) with R 2 = 0.9353, indicating that the scaling relation τ m,i ∼ Fr(z m,i ) 2 is applicable well for the majority of the fountains considered, as shown in Figure 9e.Similarly, the results for the four cases mentioned above should be excluded in the regression analysis, which produces the following correlation, with R 2 = 0.9879.This shows that the scaling relation τ m,i ∼ Fr(z m,i ) 2 is applicable very well for all fountains with the exclusion of the four fountains, as shown in Figure 9f.plotted against Fr(z m,i ) over 0.1 ≤ s ≤ 0.5 and 1 ≤ Fr ≤ 5, without exclusions, (f) τ m,i plotted against Fr(z m,i ) over 0.1 ≤ s ≤ 0.5 and 1 ≤ Fr ≤ 5, with exclusions, respectively.Figure 10 presents the numerical results for v m,a , which quantifies the increase rate of z m with time during the averaging period at the FDS. Figure 10a shows that for each s, Fr only significantly affects v m,a when Fr is no more than 3, with v m,a increasing significantly when Fr becomes larger.However, beyond Fr ≈ 3, the effect of Fr on v m,a is very small.It is also observed that the variation of s results in noticeable changes of v m,a when Fr is no more than 3, with a larger s value producing a reduced value of v m,a , but beyond Fr ≈ 3, the effect of s on v m,a becomes negligible, as shown in Figure 10b.
A multi-variable regression analysis of the results for v m,s from all fountains considered shows that Fr 0.22 s −0.155 provides a reasonable representation of the overall effects of Fr and s on v m,a , and the following correlation is obtained when all fountains are included, v m,a = −0.0011(Fr 0.22 s −0.155 ) 2 + 0.0043Fr 0.22 s −0.155 − 0.0023 = −0.0011Fr0.44 s −0.31 + 0.0043Fr 0.22 s −0.155 − 0.0023, with R 2 = 0.8036, as shown in Figure 10c.Similarly, the results of the four fountains mentioned above should be excluded in the regression analysis, and it is found that the obtained Fr 0.238 s −0.147 provides a slightly better representation of the overall effects of Fr and s on v m,a , with the following correlation obtained when the four fountains are excluded, v m,a = −0.0012(Fr 0.238 s −0.147 ) 2 + 0.0046Fr 0.238 s −0.147 − 0.0025 = −0.0012Fr0.476 s −0.294 + 0.0046Fr 0.238 s −0.147 − 0.0025, with R 2 = 0.8485, as shown in Figure 10d.

Intrusion
As explained earlier, intrusion is another key feature of the fountain behavior of an SPF in LSF.Due to the dense fluid injected, the intrusion formation can be observed to be on the bottom surface only after the downflow reaches the bottom surface and subsequently moves outward along the bottom floor.Therefore, intrusion development and its movement alter ambient fluid stratification, leading to a reduced negative buoyant force applied to the fountain.This effect is especially notable for large s and small Fr values that observed MFPH is considerably smaller.Figure 11 presents an example of the intrusion for the SPF at Fr = 2 and s = 0.1 with its temperature contours and outer boundary region.The major parameters characterizing the intrusion behavior include x int and u int , which are the instantaneous dimensionless intrusion front distance away from X = 0 and the corresponding dimensionless velocity, as depicted in Figure 11b.The numerically obtained time series of x int and u int of the SPF at Fr = 2 and s = 0.2 are presented in Figure 12.It is seen that the evolution of x int experiences three distinct stages: the initial stage (Stage 1) from the formation of the intrusion until u int attains the maximum, in which u int increases continually; subsequently, u int reduces monotonically for a period of time (Stage 2); and eventually, the intrusion is at the FDS (Stage 3) in which u int continually decreases but at the rates that are much smaller than those in Stage 2. As the instantaneous values of x int at different times are determined automatically in the code, the locations of x int , which is the furthermost point of the intrusion front, may be at different heights.These results in the fluctuations in the time series of u int due to the very small time step used and the very long whole time duration considered, as shown in Figure 12b.To smooth these fluctuations, the moving average of u int , denoted as ūint , is considered to be a better representation of u int .The time series of ūint is presented in Figure 12c.In Figure 12d, the time series of the corresponding rate of ūint changing with time, i.e., the acceleration āint , is shown.For all fountains considered, the moving average interval of 5 in dimensionless time is used.
It is noted that these three stages (Stages 1, 2 and 3) for the evolution of x int are very similar to the three regimes for the development of a pure gravity current, i.e., the wall jet regime, the buoyancy-inertial regime, and the buoyancy-viscosity regime, which are distinguished by the respective dominating forces [20].In the wall jet regime, the flow behaves as a plane wall jet with the dominant momentum, which is followed by the second regime where buoyancy becomes the driving force which is balanced by the inertial force; and gradually, the inertial force decreases and the total viscous force caused by the interfacial shear stress between the gravity current and the ambient fluid and the bottom shear stress increases, eventually evolving into the buoyancy-viscosity regime, where the buoyancy force is balanced by the viscous drag force [20].
The effects of Fr and s on x int and ūint are demonstrated by the numerical results presented in Figure 13 for five SPFs with varying Fr at s = 0.1 and five SPFs with varying s at Fr = 2.The results show that the times series of x int and ūint are very similar for all fountains, although they differ quantitatively.Fr is found to significantly affect both x int and ūint , whereas the effect of s is much smaller, particularly at the FDS (Stage 3).where u int,m is the instantaneous maximum horizontal velocity of the intrusion front and τ int,m is the time instant when u int = u int,m , ūint is the moving average of u int with the averaging period of 5 (dimensionless time), and āint is the rate of ūint changing with time (i.e., the acceleration of x int ).x int , u int , ūint , and āint are made dimensionless by X 0 , V 0 , V 0 , and X 0 /V 2 0 and V 0 , respectively.
The effects of Fr and s on ūint,m , which is the moving average of the maximum horizontal velocity of the intrusion front, are shown in Figure 14 for all SPFs considered.As shown in Figure 14a, for each s, ūint,m reduces significantly when Fr becomes larger, which can be quantified by a power-law correlation.However, the effect of s on ūint,m is negligible, as all data with different s values are essentially on the same power-law curve.This is also clearly demonstrated in Figure 14b which shows that for each Fr value, ūint,m almost does not vary, particularly when Fr is beyond 1.
Similar to that for z m , the overall effects of Fr and s on ūint,m can also be quantified by Fr a s b .A multi-variable regression analysis of the numerical results for ūint,m from all fountains considered shows that Fr −0.603 s −0.013 quantifies the overall effects of Fr and s on ūint,m very well, with the following correlation obtained when all fountains are included, ūint,m = 0.9932Fr −0.603 s −0.013 + 0.0153, ( with R 2 = 0.9925.From this correlation, it is seen that c = −0.013,showing that the effect of s is negligible, which is in agreement with the results presented in Figure 14a,b.Hence, it is expected that the omission of the effect of s on ūint,m should not lead to a noticeable change of d.This is verified by the result presented in Figure 14c, which shows that the following power-law correlation can be obtained, with R 2 = 0.9893, where a = −0.607,which is almost the same as a = −0.603as obtained in the correlation (19) when the effect of s is included.The effects of Fr and s on τ int,m , which is the time instant when ūint,m attains the maximum, are demonstrated by the numerical results presented in Figure 15.As shown in Figure 15a, for each s value, τ int,m increases significantly when Fr increases, which can be quantified by a power-law correlation.s also has an effect on τ int,m , although not as strong as Fr has, particularly when Fr is larger.This is also clearly shown in Figure 15b which shows that for each Fr value, τ int,m decreases when s increases.
Likewise, the overall effects of Fr and s on τ int,m can also be quantified by Fr a s b .A multi-variable regression analysis of the results for τ int,m from all fountains shows that Fr 0.695 s −0.14 quantifies the overall effects of Fr and s on τ int,m very well, and the following correlation is obtained when all fountains are included, τ int,m = 14.039Fr 0.695 s −0.14 − 0.2744, with R 2 = 0.9935.A similar analysis can also be conducted for āu,s1 , āu,s2 and āu,s3 , which are the time averages of the moving averages of the acceleration of the intrusion front at Stages 1, 2 and 3, respectively.It is expected that the effects of Fr and s on āu,s1 , āu,s2 and āu,s3 should also be quantified by Fr a s b , although their respective values of a and b are different, as discussed below.It should be noted that the values of āu,s1 are positive, but the values of āu,s2 and āu,s3 are negative.In the subsequent analysis of the results for āu,s2 and āu,s3 , only their magnitudes are used as their values.
The effects of Fr and s on āu,s1 are demonstrated by the numerical results presented in Figure 16.As shown in Figure 16a, for each s value, āu,s1 reduces significantly when Fr increases, which can be quantified by a power-law correlation.However, the effect of s on āu,s1 , similar to that on ūint,m , is negligible, except at Fr = 1, as all data with different s values fall approximately on the same power-law curve.This is also clearly demonstrated in Figure 16b which shows that for each Fr value, with the exception of Fr = 1, āu,s1 varies only slightly.
Similarly, the overall effects of Fr and s on āu,s1 can also be quantified by Fr a s b , as mentioned above.A multi-variable regression analysis of the results for āu,s1 from all fountains considered shows that Fr −1.291 s −0.024 quantifies the combined effects of Fr and s on āu,s1 reasonably well, and the following correlation is obtained when all fountains are included, āu,s1 = 0.03154Fr −1.291 s −0.024 + 0.00197, with R 2 = 0.9176.From this correlation, it is seen that b = −0.024,indicating that the effect of s is negligible, which is in agreement with the results presented in Figure 16a,b.Hence, it is expected that the omission of the effect of s on āu,s1 should not lead to a noticeable change of a.This is verified by the result presented in Figure 16c, which shows that the following power-law correlation can be obtained, āu,s1 = 0.0385Fr −1.299 , ( with R 2 = 0.9362, where a = −1.299,which is almost the same as a = −1.291as obtained in the correlation (22) when the effect of s is included.It should be noted that the relatively large deviations of some data, particularly those at Fr = 1, away from the correlations ( 22) and (23), are due to two reasons.The first one is because for all fountains considered, the moving average interval of 5 (dimensionless time) is used.This leads to the absence of the data within the initial 5 in the determination of the moving average of u int , i.e., ūint , in Stage 1, which in turn excludes these data for āu,s1 .As in general, the durations of Stage 1 are relatively small, from about 16 at Fr = 1 to the maximum of about 50 at Fr = 5 for all fountains considered, the absence of the data within the initial 5 in the determination of the moving average of u int in Stage 1 has a significant effect, particularly at Fr = 1 and other small Fr values.The second reason is that it is observed that the changes of ūint are substantial at different times within Stage 1, particularly at small Fr values, which results in a relatively large inaccuracy of the values obtained for āu,s1 .However, as the changes of ūint within Stage 2 and Stage 3 are much smaller and additionally, there is no issue with the absence of the data within the initial 5 in the determination of the moving average of u int within these two stages, it is expected that the values of d and c in Fr a s b for āu,s2 and āu,s3 , as well as the relevant correlations are more accurate, as shown below.The effects of Fr and s on āu,s2 are demonstrated by the numerical results presented in Figure 17.As shown in Figure 17a, for each s value, āu,s2 reduces significantly when Fr increases, which can be quantified by a power-law correlation.It is also seen that s affects āu,s2 , with its increase leading to a larger āu,s2 value for each Fr value, as demonstrated in Figure 17b, but the effect of s on āu,s2 is significant when Fr = 1, and it is much smaller for higher Fr values.A multi-variable regression analysis of the results for āu,s2 from all fountains considered shows that Fr −2.074 s 0.615 quantifies the overall effects of Fr and s on āu,s2 well, and the following correlation is obtained from the numerical results of all fountains, āu,s2 = 0.04416Fr −2.074 s 0.615 + 0.000021, with R 2 = 0.9742.The effects of Fr and s on āu,s3 are demonstrated by the numerical results presented in Figure 18.As shown in Figure 18a, for each s value, āu,s3 reduces significantly when Fr increases, which can be quantified by a power-law correlation.It is also seen that s affects āu,s2 , with its increase leading to a larger āu,s3 value for each Fr value, as demonstrated in Figure 18b, and similar to that for āu,s2 , the effect of s on āu,s3 is larger when Fr is smaller.A multi-variable regression analysis of the results for āu,s3 from all fountains considered shows that Fr −0.683 s 0.35 quantifies the overall effects of Fr and s on āu,s3 relatively well, and the following correlation is obtained when all fountains are included, āu,s3 = 0.000783Fr −0.683 s 0.35 − 0.000001, with R 2 = 0.9482.

Conclusions
The characterization of weak plane fountains in stratified fluids with small Fr and Re values, in which only the symmetric behavior is present, is currently rarely understood.Furthermore, the intrusion is an integral part of a weak symmetric plane fountain, and its behavior is important due to its significant effects on the maximum fountain penetration height.Nonetheless, intrusion in such weak symmetric plane fountains, particularly in stratified ambient fluids, is rarely investigated.A numerical study was thus conducted on the weak symmetric plane fountains in linearly stratified fluids with simulations over 0.1 ≤ s ≤ 0.5 and 1 ≤ Fr ≤ 5, all at constant Re = 100.The parameters selected to investigate the fountain behavior are z m , both initial and time-averaged, the time to attain the initial z m , along with as the velocities of intrusion at different stages.
There are two major differences in behavior between a weak SPF in a homogeneous fluid and that in an LSF.One difference is that the stratification of the ambient fluid stabilizes the symmetry of the weak fountain, which makes the fountain become asymmetric at a larger Fr value.The other difference is that z m of the SPF in the LSF continues to increase at the FDS, whereas that in the homogeneous fluid is essentially constant.The observed continuous increase of the MFPH in the SPF is largely caused by the intrusion which continuously diminishes with time at the FDS, leading to reduced buoyant force experienced by the fountain fluid.This is especially notable for large s and small Fr values with the MFPH considerably smaller.
The results show that z m and the associated time are under the effects of Fr and s, with the effect of Fr usually stronger than that of s.The overall effects of Fr and s can be quantified by Fr a s b , with the values of a and b varying for different parameters related to z m .With the numerical results obtained for all weak SPFs, empirical correlations are produced in terms of Fr a s b for each relevant parameter, which generally predict the results very well.It is further found that if the local Froude number Fr(z m,i ), which incorporates the effect of s, is used instead of Fr, the scaling relations using Fr(z m,i ) only developed by Lin and Armfield [6] for weak SPFs in LSFs basically also work well for the SPFs considered in the present study.
The evolution of the intrusion experiences three distinct stages; Stage 1 is the initial stage, from the formation of the intrusion until u int attains the maximum, in which u int increases continually; this is followed by Stage 2 in which u int reduces monotonically for a period of time; and eventually the intrusion is in Stage 3, which is at the FDS in which u int continually decreases but at rates that are much smaller than those in Stage 2. The results show that both Fr and s have effects on u int and the associated rates of changes with time (accelerations) at different stages and similarly the overall effects of Fr and s on these parameters can also be quantified in terms of Fr a s b , with different values of of a and b.Empirical correlations are obtained in terms of Fr a s b for each relevant parameter, which generally predict the results well.

Figure 1 .
Figure 1.Evolution of the transient temperature contours of the PF at Fr = 3.25 and s = 0.1 at Y = 0 in the X-Z plane (first column), X = 0 in the Y-Z plane (second column), and z = 0.5z m,i in the X-Y plane (third column).Temperatures are non-dimensionalized with [T(Z) − T 0 ]/(T a,z=60 − T 0 ).

Figure 2 .
Figure 2. Snapshots of the temperature contours at the FDS of SPFs for several Fr values with s = 0.2, at Y = 0 in the X-Z plane (first column), X = 0 in the Y-Z plane (second column), and z = 0.5z m,i in the X-Y plane (third column), respectively.

Figure 3 .
Figure 3. Snapshots of the temperature contours at the FDS of SPFs for several s values with Fr = 2, at Y = 0 in the X-Z plane (first column), X = 0 in the Y-Z plane (second column), and z = 0.5z m,i in the X-Y plane (third column).

Figure 4 .
Figure 4. Time series of z m (a) and v m (b) for the SPF of Fr = 2 and s = 0.1, where z m,i is the initial MFPH and τ m,i is the time instant when z m = z m,i , z m,s is the MFPH at the commencement time of the period for the time averaging of z m at the FDS which gives the time-averaged MFPH z m,a .All heights, times and velocity are made dimensionless by X 0 , X 0 /W 0 and W 0 , respectively.The averaging period for both z m,a and v m,a is 100 ≤ τ ≤ 900.

Figure 5 .
Figure 5.Time series of z m (a) and v m (c) for the five SPFs with five Fr values at s = 0.1, and that of z m (b) and v m (d) for the five SPFs with five s values at Fr = 2, respectively.

Figure 11 .
Figure 11.(a) The temperature contour and (b) the outer boundary of the intrusion region at Y = 0 in the X-Z plane, which is the iso-temperature curve at T(Z) = T 0 − 1%(T a,0 − T 0 ), for the SPF at Fr = 2 and s = 0.1, where x int and u int are the instantaneous intrusion front distance away from X = 0 and the corresponding velocity, which are made dimensionless by X 0 and W 0 , respectively.

Figure 12 .
Figure12.Time series of (a) x int , (b) u int , (c) ūint , and (d) āint for the SPF at Fr = 2 and s = 0.1, where u int,m is the instantaneous maximum horizontal velocity of the intrusion front and τ int,m is the time instant when u int = u int,m , ūint is the moving average of u int with the averaging period of 5 (dimensionless time), and āint is the rate of ūint changing with time (i.e., the acceleration of x int ).x int , u int , ūint , and āint are made dimensionless by X 0 , V 0 , V 0 , and X 0 /V 2 0 and V 0 , respectively.

Figure 13 .
Figure 13.Time series of x int (a) and ūint (c) for the five SPFs with five Fr values at s = 0.1, and the time series of x int (b) and ūint (d) for the five SPFs for the five s values at Fr = 2, respectively.