On the thermal instability of supercavitating liquid jet surrounded by coaxial rotary gas

Based on the jet stability theory, under the conditions of gas rotation, fluid compressibility and supercavitation, this paper gives the mathematical model describing the thermal instability of supercavitating liquid jet surrounded by a coaxial rotary gas, and the corresponding numerical method for solving the mathematical model is proposed and verified by the data in reference. Then, this paper analyzes the effects of gas–liquid temperature differences and temperature gradients on jet instability, and studies the thermal stability of supercavitating jet. The results show that the maximum disturbance growth rate, the dominant frequency and the maximum disturbance wave numbers increase linearly with the increase of gas–liquid temperature differences. The existence of temperature gradient inside the jet makes the effects of temperature differences on jet instability more obvious. The temperature gradient will inhibit the effect of supercavitation on jet instability, while gas–liquid temperature difference will promote the effect of supercavitation on jet instability.


INTRODUCTION
In some engineering applications, such as fuel injection process of an internal combustion engine, the gas temperature in the cylinder is far higher than the fuel jet temperature, and there may be a certain temperature gradient inside the jet liquid; the instability of the jet is inevitably affected by the thermal effect. Study on the jet thermal instability is the deeper discussion on jet stability, which can further promote the understanding of liquid jet splitting and atomization, and also have an important academic significance and engineering application value.
Supercavitating liquid jet in the fuel injection process refers to the kind of phenomenon that cavitation bubbles exist in the fuel jet leaving the nozzle. With the constant rise of the fuel injection pressure, supercavitating liquid jet has become very prominent.
To our knowledge, there is no uniform and universally recognized definition of the thermal instability of the liquid jet at present. Most scholars gave related thermal instability definitions based on corresponding research content. For example, Xu and Davis [1] considered temperature gradient existing along the axial direction of the jet as a thermal instability problem, while Ding [2] considered temperature difference existing between the jet surface and surrounding gas as a thermal instability problem. In this paper, based on the historical definition [1][2][3][4] of thermal instability problem, we defined jet thermal instability as jet instability caused directly or indirectly by thermal effect.
With the deep research on thermal instability of liquid jet, at present, some scholars already provided some mathematical models under different conditions based on linear stability theory to describe the stability of liquid jet, and also obtained many important theoretical research achievements [5][6][7][8][9]. However, these mathematical models and the corresponding research results are often obtained by neglecting the thermal effect. Due to the complexity, some studies on thermal stability of jet are carried out under specified simplified conditions [1,2,10,11]; for example, it is assumed that the temperature disturbance is axisymmetric, the jet is simplified as plane stratified flow and so on. At present, the research on the thermal stability of liquid jet is still at the exploratory stage; it is not definite and clear of the mechanism of liquid jet breakup and atomization caused directly or indirectly by thermal effect. For diesel jet, with the increase of fuel injection pressure, the effect of compressibility of the surrounding gas on jet stability is more and more obvious [6,12]. In addition, fuel injection is always accompanied with rotating motion of airflow in cylinder [13,14]. Study on the thermal instability of the liquid jet in a coaxial rotating compressible gas has not yet been reported.
Considering current study on thermal stability of liquid jet, two aspects will be studied in this paper: (1) jet thermal instability caused by the development of the disturbance wave on the jet surface that is dominated by the temperature gradient inside the jet and (2) jet thermal instability caused by the thermal disturbance that is dominated by the temperature difference between liquid jet surface and surrounding gas. Based on the definition of thermal instability, the thermal instability of liquid jet studied in this paper belongs to the scope of thermal instability problem caused indirectly by thermal effect. Based on the previous research work [15][16][17], under the conditions of gas rotation, fluid compressibility and supercavitation, this paper gives the mathematical model describing the thermal instability of supercavitating jet surrounded by a coaxial rotary gas, and the corresponding numerical method for solving the mathematical model is proposed and verified by the data in reference. Also, the effects of gas-liquid temperature differences and temperature gradients on jet instability are analyzed, respectively, and then the thermal instability of supercavitating liquid jet is studied.

MATHE MATICAL MODEL 2.1 Physical model and initial flow field
Suppose that supercavitating liquid jet is injected into the coaxial rotary gas. Jet initial temperature is T 1 (r) (the central temperature is always T 1 ), jet initial radius is a, jet initial speed is U 0 , jet density is ρ 1 , the surrounding gas temperature is T 2 , gas has no velocity in the direction of z-axis, rotational strength is W 0 and gas density is ρ 2 . The initial disturbance of the jet interface is η 0 . The column coordinates (r, θ, z) are set up at the outlet of the jet nozzle, and jet direction is opposite to the z-axis, as shown in Fig. 1.
Based on the above physical model, some assumptions are indicated as follows [15][16][17]: 1. The initial temperature distribution in liquid jet is linear along the r-direction. Jet central temperature is T 1 , and remains unchanged. Jet surface temperature is T 2 . 2. Both liquid jet and ambient gas are compressible Newtonian fluids. 3. Neglect the effects of viscosity and gravity of liquid jet and surrounding gas. 4. The surrounding gas is a coaxial rotary flow, while liquid jet is not rotary. 5. The mixed phase made up of uniformly distributed cavitation bubbles and liquid jet is regarded as continuous medium.
In the given coordinate system, the basic flow field of an undisturbed jet (liquid jet temperature and velocity, external gas velocity, the pressure difference between external gas and liquid jet, jet density and the sound velocity in the jet) can be expressed as follows [15][16][17][18][19]: where V 1 is the jet velocity, V 2 is the surrounding gas velocity, p 1 is the jet pressure, p 2 is the surrounding gas pressure, σ 0 is the surface tension coefficient, ρ v is the density of cavitation bubbles, ρ l is the liquid density, c 1 is the sound velocity in the jet, α is the bubble volume fraction and a bigger α means a stronger cavitation phenomenon, k v is the adiabatic index of cavitation bubbles, z v is the compression factor of cavitation bubbles, R is the molar gas constant and E l is the liquid elastic modulus. The horizontal line on the symbol of physical quality represents the undisturbed physical quantity. In addition, Eqs (5) and (6) are used to model cavitation phenomenon.

The establishment and solution of the mathematical model
The research subjects of this paper (the jet and surrounding gas) satisfy mass, momentum and energy conservation equations as follows [20,21]: where i = 1 represents jet parameters, i = 2 represents surrounding gas parameters, λ i is the thermal conductivity and cp i is the constant pressure specific heat. We conduct perturbation and linearized analysis on Eqs (7)- (9), and then get the following governing equations of jet disturbance. For the liquid jet: For the surrounding gas: where apostrophe represents disturbance parameters.
At the interface between the liquid jet and the surrounding gas, the corresponding boundary conditions (including kinematic, dynamic, temperature continuity and heat flux continuity conditions) are listed as follows [2,[15][16][17]22]: where η is the disturbance of the jet interface, r 1 is the principal curvature radius of liquid at the interface and r 2 is the principal curvature radius of gas at the interface.
Conducting the perturbation analysis on Eqs (16)- (19), we get the perturbation expressions: Conducting the linearized analysis (deleting the second-order small quantity) on Eq. (20) and combining Eqs (16)- (19), we get the following boundary conditions: Uniting the above jet disturbance governing equations, boundary conditions and the undisturbed basic flow field of jet, we can obtain a mathematical model for describing the thermal instability of a supercavitating jet surrounded by a compressible rotary gas. The mathematical model is characterized by 13 independent parameters. Thus, this mathematical model can be expressed as follows: where k = k r + ik i , k r is the wave number in the direction of Z, k i is the spatial growth rate of disturbance, ω = ω r + iω i , ω r is the temporal growth rate of disturbance, ω i is the wave frequency, m is the angular modulus, We = σ /ρ 1 U 2 0 a is the reciprocal of the Weber number, E = W 0 /(U 0 a) is the nondimensional rotational strength of surrounding gas, Ma 1 = U 0 /c 1 is the liquid Mach number, Ma 2 = U 0 /c 2 is the gas Maher number, Q = ρ 2 /ρ 1 is the density ratio of gas and liquid, Pe 1 = ρ 1 c p1 U 0 a/λ 1 is the Peclet number of liquid jet, Pe 2 = ρ 2 c p2 U 0 a/λ 2 is the Peclet number of surrounding gas, k 0 = c p2 /c v2 is the ratio of constant pressure specific heat and constant volume specific heat, λ n = λ 2 /λ 1 is the ratio of gas thermal conductivity and liquid thermal conductivity and T r = ( T 2 − T 1 )/T 2 is the ratio of internal temperature difference in jet and the gas temperature.
Both real and imaginary parts of Eq. (22)  Based on the mathematical model of Eq. (22), the left-hand side of the equation is a complex function that contains 13 variables. T r can not only reflect the effect of temperature gradient inside the jet on the jet instability, but also reflect the effect of thermal disturbance caused by temperature difference between liquid and gas on the jet instability.
Equation (22) needs to be solved iteratively. Researchers used to adopt the parabola method (Müller method) [12,13] to solve this kind of mathematical model. Compared with the secant method [23], the parabola method has better convergence rate, but its algorithm is more complex. The symbol problem and complex operation need to be considered when using the parabola method. Therefore, in this paper, the secant method is used to solve the mathematical model iteratively. Figure 2 shows the schematic of numerical solving process, and the numerical calculation process is implemented by Mathematica software.

Verification of the mathematical model
Suppose that there is only axisymmetric disturbance at the jet surface, no linear temperature distribution and no cavitation bubbles in the liquid jet, the surrounding gas is stationary and the fluids are incompressible, which means m = 0, T r = 0, α = 0, E = 0 and Ma 1 = Ma 2 = 0; thus, Eq. (22) is reduced to Equation (23) agrees with the mathematical model of Lin and Lian [24] in their studies of spatial instability of inviscid liquid jet under the condition of axisymmetric disturbance. So, the mathematical model established in this paper is correct to some extent.
To validate the mathematical model and its numerical solving method, we calculated by adopting parameters in the literature [25], and compared with the results provided in this literature. Figure 3 shows the comparison results. Schmidt et al. [25] gave a case of an inviscid incompressible liquid jet into an inviscid incompressible stationary gas under the condition of axisymmetric disturbance. As shown in Fig. 3, the calculation results in this paper are consistent with the reference data in the literature, which indicates that the numerical solution method is also reasonable and effective. Figure 4 provides the comparison of the disturbance growth rate versus wave number with and without temperature difference between jet surface and surrounding gas when there is no linear temperature distribution in the radial liquid jet.  T 2 / T 2 is the ratio of surrounding gas temperature to jet surface temperature. It can be used to characterize the temperature difference between the jet surface and surrounding gas. T 2 / T 2 = 1 means no temperature difference between jet surface and surrounding gas, while T 2 / T 2 > 1 means that surrounding gas temperature is higher than jet surface temperature.
As Fig. 4 shows, the disturbance growth rate on the jet surface significantly increases when temperature difference exists compared with the value without temperature difference, which indicates that the temperature difference between the jet surface and surrounding gas will expedite the instability of the liquid jet, and also favor the jet breakup and atomization. This conclusion is consistent with the experiment result provided by the literature [26] that studied the effect of surrounding gas temperature on atomization of diesel jet.
According to the above comparison from different angles, we can approve the correctness of mathematical model and its numerical solution method.

RESULTS AND DISCUSSION
Diesel jet is chosen as a research object. Calculation parameters are set according to [15,[26][27][28], which are shown in Table 1.

Effects of gas-liquid temperature difference on jet instability
The temperature difference between liquid jet surface and surrounding gas is the major reason of thermal disturbance on the jet surface, which has an important influence on the jet instability. The effect of gas-liquid temperature difference on jet instability will be analyzed below; meanwhile, the temperature gradient inside the jet is not considered for now. Note that T 2 / T 2 is the ratio of surrounding gas temperature to jet surface temperature. It is used to characterize the temperature difference between the surrounding gas and liquid jet surface.   Figure 5 presents the comparison of the variation of disturbance growth rate versus axial wave numbers with and without gas-liquid temperature difference between the surrounding gas and liquid jet surface, when angular modulus m = 0 (axisymmetric disturbance) and m = 1 (asymmetric disturbance). T 2 / T 2 = 1 and 1.17 represent gas-liquid temperature difference is 0 and 50 K, respectively.
As shown in Fig. 5, when gas-liquid temperature difference exists (T 2 / T 2 = 1.17), the disturbance growth rates under different wave numbers are obviously greater than those when T 2 / T 2 = 1, the maximum disturbance growth rate corresponding to axial wave number increases, and the range of the axial wave number is also obviously widened. The diameter of the atomized droplet can be calculated by the conservation of volume that requires the relation between the radius of droplet R and the wavelength of disturbance λ, πa 2 λ = 4πR 3 /3, where λ is related to the wave number k r by k r = 2πa/λ, which indicates that the diameter of the atomized droplet R is inversely proportional to axial wave number k r . Therefore, thermal disturbance caused by gas-liquid temperature difference will destabilize the liquid jet, the droplet size will reduce, the quantity of small droplets will increase and the size range of droplets will become wider.
In addition, comparing Fig. 5a and b, it is also found that the effect of gas-liquid temperature difference on axisymmetric disturbance is similar to the effect of gas-liquid temperature difference on asymmetric disturbance, which indicates that the effect of thermal disturbance caused by gas-liquid temperature difference on jet instability has little relationship with the disturbance wave types.
Note that the maximum disturbance growth rate k i_max , dominant frequency ω i_max (the frequency corresponding to the maximum disturbance growth rate) and the maximum disturbance wave number k r_max are the three most important parameters to characterize the jet instability. Figure 6 provides the variations of the maximum disturbance growth rate, dominant frequency and the maximum disturbance wave number under different gas-liquid temperature differences. T 2 / T 2 = 1, 1.17 and 1.33 represent gas-liquid temperature difference is 0, 50 and 100 K, respectively.
As shown in Fig. 6, with the increase of gas-liquid temperature differences, the maximum disturbance growth rate, dominant frequency and the maximum disturbance wave number show an approximately linear increasing trend. It means that thermal disturbance caused by gas-liquid temperature difference has an obvious impact on prompting jet instability. Also, according to πa 2 λ = 4πR 3 /3 and k r = 2πa/λ, and referring to Fig. 6c, when T 2 / T 2 = 1, the maximum disturbance wave number k r_max is 163, so the smallest atomized droplet radius R is ∼30.7 μm. When T 2 / T 2 = 1.33, the maximum disturbance wave number k r_max is 200, so the small- est atomized droplet radius R is ∼28.7 μm. Therefore, when gas-liquid temperature difference is taken into account, the size of the smallest atomized droplet will reduce, and the size range of droplets will become wider.

Effects of temperature gradient on jet instability
Section 3.1 analyzed the effect of gas-liquid temperature difference on jet instability with the neglect of the temperature gradient inside the jet (T r = 0, jet central temperature T 1 equals jet surface temperature T 2 ). Next, the effect of gas-liquid temperature difference on jet instability considering the temperature gradient inside the jet and the effect of the temperature gradient inside the jet on jet instability will be studied. Note that T r = 0 means no temperature gradient inside the jet, and T r = 0.14 means temperature gradient inside the jet exists. Figure 7 shows the comparison of the disturbance growth rates under different gas-liquid temperature differences considering the temperature gradient inside the jet (T r = 0.14). T 2 / T 2 = 1, 1.14 and 1.29 represent that gas-liquid temperature difference is 0, 50 and 100 K, respectively.
As Fig. 7 shows, when temperature gradient inside the jet exists (T r = 0.14), with the increase of gas-liquid temperature differences, the disturbance growth rate, the maximum disturbance growth rate corresponding to axial wave number and the range of the axial wave number increase. In addition, comparing Fig. 7a and b, it is also found that the effect of gas-liquid temperature difference on axisymmetric disturbance is similar to that on asymmetric disturbance when temperature gradient exists. This trend is the same as that without considering the temperature gradient. However, comparing Figs 5 and 7, when gas-liquid temperature difference is 50 K, the effect of gas-liquid temperature difference on jet instability considering the temperature gradient is greater than that without considering the temperature gradient.
To further study the effect of gas-liquid temperature difference on jet instability considering the temperature gradient inside the jet, Fig. 8 provides the comparison of the effect of gas-liquid temperature difference on jet instability with and without temperature gradient inside the jet.
As shown in Fig. 8, whether temperature gradient inside the jet exists or not, the maximum disturbance growth rate, dominant frequency and the maximum disturbance wave number show an approximately linear increasing trend with the increase of gas-liquid temperature differences. However, compared with no temperature gradient inside the jet, the increments of maximum disturbance growth rate, dominant frequency and maximum disturbance wave number considering the temperature gradient will increase with the increase of gas-liquid temperature differences. Take the maximum disturbance growth rate for example. If temperature gradient inside the jet does not exist (T r = 0), the maximum disturbance growth rate is 2.27 when gas-liquid temperature difference is 0 K, while the maximum disturbance growth rate is 2.78 when gas-liquid temperature difference is 100 K, and the maximum disturbance growth rate increased 22.47%. If temperature gradient inside the jet exists (T r = 1.4), the maximum disturbance growth rate is 2.78 when gas-liquid temperature difference is 0 K, while the maximum disturbance growth rate is 3.60 when gas-liquid temperature difference is 100 K, and the maximum disturbance growth rate increased 29.50%. Therefore, the existence of temperature gradient inside the jet makes the effects of temperature differences on jet instability more obvious.

Thermal instability of supercavitating jet
Some research results show that supercavitation has an important effect on the stability of liquid jet [17,18]; also Section 3.1 and 3.2 analyzed the effect of gas-liquid temperature difference and temperature gradient on the stability of liquid jet. Next, the effect of temperature gradient inside the jet and gas-liquid temperature difference on supercavitating jet instability will be studied. Figure 9 presents the comparison of the disturbance growth rates under different bubble volume fractions with and without temperature gradient inside the jet. T r = 0 represents no temperature gradient inside the jet, and T r = 0.25 represents temperature gradient inside the jet exists (jet surface temperature T 2 is 100 K higher than jet central temperature T 1 ).
As shown in Fig. 9, whether temperature gradient inside the jet exists or not, the disturbance growth rates under different wave numbers increase with the increase of bubble volume fractions, which indicates that supercavitation (bubble volume fraction) has a positive impact on the jet instability. However, comparing Fig. 9a and b, it is found that the effect of supercavitation on the maximum disturbance growth rate when temperature gradient exists is obviously smaller than that when there is no temperature gradient inside the jet. As Fig. 9 shows, if there is no temperature gradient inside the jet, the maximum disturbance growth rate is 2.23 when bubble volume fraction is 0, while the maximum disturbance growth rate is 2.59 when bubble volume fraction is 0.1, and the maximum disturbance growth rate increased 16.14%. If temperature gradient exists inside the jet, the maximum disturbance growth rate is 3.57 when bubble volume fraction is 0, while the maximum disturbance growth rate is 3.70 when bubble volume fraction is 0.1, and the maximum disturbance growth rate increased 3.64%. So, the temperature gradient inside the jet will inhibit the effect of supercavitation on the maximum disturbance growth rate.
Besides, comparing Fig. 9a and b, it is also found that the effect of supercavitation on the maximum disturbance wave number when temperature gradient exists is obviously smaller than that when there is no temperature gradient inside the jet, which means that the temperature gradient inside the jet will also inhibit the effect of supercavitation on the maximum disturbance wave number, and  further inhibit the effect of supercavitation on the size range of atomized droplets. Therefore, the temperature gradient will inhibit the effect of supercavitation on jet instability. To further study the effect of temperature gradient on supercavitating jet instability, Fig. 10 provides the comparison of the effect of supercavitation on the jet instability with and without temperature gradient inside the jet. T r = 0 and 0.25 represent T 2 − T 1 = 0 and 100 K, respectively.
As Fig. 10a shows, with the increase of bubble volume fractions, the maximum disturbance growth rates increase, but this trend is more obvious when T r = 0 than T r = 0.25. It is proved again that temperature gradient inside the jet can inhibit the effect of bubble volume fraction on jet instability.
As shown in Fig. 10b and c, when T r = 0, both the dominant frequency and the maximum disturbance wave number significantly increase with the increase of bubble volume fractions, while both of them show a special change (first descend and then ascend) with the increase of bubble volume fractions when T r = 0.25. It indicates that temperature gradient inside the jet may change the effect of supercavitation on jet instability when bubble volume fraction is small and can effectively inhibit the effect of supercavitation on jet instability even when the value of bubble volume fraction gets greater. Figure 11 presents the comparison of the disturbance growth rates under different bubble volume fractions with and without gasliquid temperature difference between the surrounding gas and liquid jet surface. T 2 / T 2 = 1 and 1.33 represent that gas-liquid temperature difference is 0 and 100 K, respectively. As shown in Fig. 11, whether gas-liquid temperature difference between the surrounding gas and liquid jet surface exists or not, the disturbance growth rates significantly increase with the increase of bubble volume fractions. Comparing Fig. 11a and b, it is seen that the disturbance growth rates under different bubble volume fractions when T 2 / T 2 = 1.33 are obviously greater than those when T 2 / T 2 = 1.
In addition, comparing Fig. 11a and b, it is also found that the effect of supercavitation on the disturbance growth rate when T 2 / T 2 = 1.33 is similar to that when T 2 / T 2 = 1, which is absolutely different from the effect of temperature gradient inside the jet. Figure 12 provides the comparison of the effect of supercavitation on the jet instability with and without gas-liquid temperature difference between the surrounding gas and liquid jet surface. T 2 / T 2 = 1 and 1.33 represent that gas-liquid temperature difference is 0 and 100 K, respectively.
As shown in Fig. 12, whether gas-liquid temperature difference between the surrounding gas and liquid jet surface exists or not, the maximum disturbance growth rate, dominant frequency and the maximum disturbance wave number increase with the increase of bubble volume fractions. In addition, compared with T 2 / T 2 = 1, the increments of the maximum disturbance growth rate, dominant frequency and the maximum disturbance wave number ascend with the increase of bubble volume fractions when T 2 / T 2 = 1.33, which indicates that gas-liquid temperature difference will promote the effect of supercavitation on jet instability, and the promoting effect increases with the increase of supercavitation degree of liquid jet. In general, the promoting effect is not very obvious.

CONCLUSIONS
1. Under the conditions of gas rotation, fluid compressibility and supercavitation, the mathematical model describing the thermal instability of supercavitating jet surrounded by a coaxial rotary gas is given, and the corresponding numerical method for solving the mathematical model is proposed and verified. 2. When gas-liquid temperature difference exists, the disturbance growth rate is obviously greater than that when there is no gas-liquid temperature difference between the surrounding gas and liquid jet surface. The maximum disturbance growth rate, dominant frequency and the maximum disturbance wave number show an approximately linear increasing trend with the increase of gas-liquid temperature differences. The effect of thermal disturbance caused by gas-liquid temperature difference on jet instability has little relationship with the disturbance wave types. 3. Compared with no temperature gradient inside the jet, the increments of maximum disturbance growth rate, dominant frequency and maximum disturbance wave number considering the temperature gradient will increase with the increase of gasliquid temperature differences. The existence of temperature gradient inside the jet makes the effects of temperature differences on jet instability more obvious. 4. Temperature gradient inside the jet will inhibit the effect of supercavitation on jet instability. Gas-liquid temperature difference will promote the effect of supercavitation on jet instability, and the promoting effect increases with the increase of supercavitation degree of liquid jet. In general, the promoting effect is not very obvious.
NOMENCL ATURE a = nozzle radius, m E = dimensionless rotational strength k 0 = the ratio of constant pressure specific heat and constant volume specific heat k i = dimensionless spatial disturbance growth rate k i_ max = dimensionless maximum spatial disturbance growth rate k r = dimensionless wave numbers in the jet direction k r_ max = dimensionless maximum wave numbers in the jet direction m = dimensionless angular modulus Ma 1 = liquid Mach number Ma 2 = gas Maher number p 1 = jet pressure, Pa p 2 = gas pressure, Pa Pe 1 = jet Peclet number Pe 2 = gas Peclet number Q = gas-liquid density ratio T 1 = the jet temperature, K T 2 = the surrounding gas temperature, K T r = the ratio of internal temperature difference in jet and the gas temperature V 1 = jet velocity, m/s V 2 = gas velocity, m/s We = Weber number w i = dimensionless frequency of oscillation w i_max = dimensionless dominant characteristic wave frequency α = the bubble volume fraction η = the disturbance of the jet interface η 0 = the initial disturbance of the jet interface λ i = the thermal conductivity ρ 1 = jet density, kg/m 3 ρ 2 = gas density, kg/m 3 σ = surface tension factor, N/m

DATA AVAIL ABILIT Y
The data that support the findings of this study are available from the corresponding author upon reasonable request.

ACKNOWLEDGE MENTS
This project was supported by the Fundamental Research Funds for the Central Universities (grant no. 2020JBM053) and the National Natural Science Foundation of China (grant no. 51776016).