Transient Energy Growth in a Free Cylindrical Liquid Jet

Abstract

The stability and behavior of jet flows are critical in various engineering applications, yet many aspects remain insufficiently understood. Previous studies predominantly relied on modal methods to describe small perturbations on jet flow surfaces through the linear superposition of modal waves. However, these approaches largely neglected the interaction between different modes, which can lead to transient energy growth and significantly impact jet stability. This work addresses this gap by focusing on the transient growth of disturbances in jet flows through a comprehensive non-modal analysis, which captures the short-term energy evolution. Unlike modal analysis, which provides insights into the overall trend of energy changes over longer periods, non-modal analysis reveals the instantaneous dynamics of the disturbance energy. This approach enables the identification of transient growth mechanisms that are otherwise undetectable using modal methods, which treat disturbance waves as independent and fail to account for their coupling effects. The results demonstrate that non-modal analysis effectively quantifies the interplay between disturbance waves, capturing the nonlinearity inherent in transient energy growth. This method highlights the short-term amplification of disturbances, providing a more accurate understanding of jet flow stability. Furthermore, the impact of dimensionless parameters such as the Reynolds number, Weber number, and initial wave number on transient energy growth is systematically analyzed. Key findings reveal the optimal conditions for maximizing energy growth and elucidate the mechanisms driving these phenomena. By integrating non-modal analysis, this study advances the theoretical framework of transient energy growth, offering new insights into jet flow stability and paving the way for practical improvements in fluid dynamic systems.

1. Introduction

Jet flow phenomenon is widely present in people’s production and life. The rupture of the fuel jet is essentially the process of the development of small-amplitude disturbances on the surface of the jet until it breaks, that is, the process of the jet “instability”. In order to describe this process, researchers have established various mathematical models. Lord Rayleigh [1] first employed linear stability analysis to investigate the temporal stability of inviscid liquid jets. Chandrasekhar [2] considered the effect of viscosity, while Keller et al. [3] studied the spatial instability of jet flows. Lin et al. [4] considered the impact of gas viscosity. Yang et al. [5] and Liu [6] studied the instability of non-Newtonian liquid jets. Yang et al. [7] also investigated the weakly nonlinear stability using theoretical and numerical simulation methods. Additionally, Yang et al. [8] and Si et al. [9] considered the effects of electrical fields and thermal effects, respectively. Research on the changes in jet energy is crucial for understanding jet transitions, atomization, and jet breakup. The previous studies all employed modal methods to describe the small perturbations on the surface of the jet flow, which represent the linear superposition of different modal waves. However, the interaction between different modes leading to transient energy growth was not considered, which can also affect the stability of the jet flow.

Transient energy growth is an intriguing phenomenon in fluid mechanics that has received much attention in recent years. It is a process in which a small perturbation in a laminar flow, instead of decaying, grows exponentially for a short time before eventually decaying. During the combustion process of a rocket engine, the fuel jet may undergo premature breakup due to transient growth, which can significantly impact the subsequent atomization and combustion processes. The mechanism behind this phenomenon is due to the non-normality of the linearized operator governing the flow, which leads to non-orthogonal eigenvectors and, therefore, transient energy growth [10]. Transient energy growth has important implications in various fluid systems, such as pipe flows, channel flows, boundary layers, and jets. In particular, it is closely related to the flow stability and transition to turbulence, which is one of the most challenging problems in fluid mechanics. According to classical theories, laminar flow transitions to turbulent flow below the critical Reynolds number and above the Reynolds number condition for energy growth [11,12,13,14,15,16,17]. The phenomenon of the shear flow’s transition was first discovered by Patel and Head [18] and later developed by Lundbladh et al. [19] and Tillmark et al. [20]. Since then, researchers have increasingly focused on the phenomenon of transient energy growth, especially in 2D- and 3D-plane Poiseuille and Couette flows, as well as the transient response to external excitations. Significant contributions have been made by Farrell [21], Gustavsson [22,23], and Trefethen et al. [24,25,26]. In particular, they have studied the transient growth of disturbances in the linearized Navier–Stokes equations and developed efficient numerical methods to calculate and analyze the transient energy growth. These studies provide insights into the mechanisms of flow transition and help to explain the behavior of shear flow in the presence of external disturbances, which is important for practical applications such as flow control and drag reduction.

The study of transient energy growth has led to the development of non-modal analysis, which is an effective tool for understanding the transient behavior of fluid flows. Unlike the traditional modal analysis that focuses on the eigenmodes of the linearized governing equations, non-modal analysis considers the non-normality of the linear operators and the short-term transient growth of disturbances. Non-modal analysis has been shown to be a powerful tool for understanding the underlying mechanisms of flow transition and predicting the onset of turbulence, as well as for optimizing control strategies and designing flow control devices. A comprehensive review of this field can be found in the works of Reddy and Henningson [27] and Schmid [28]. Reddy and Henningson presented the theory of non-modal analysis and its application to various flow systems, while Schmid focused on the numerical methods used to study transient growth and the influence of parameters on the results. These works provide a solid foundation for the further exploration of transient energy growth and its applications in fluid mechanics.

In recent studies, researchers have realized that, in addition to small disturbances, transient energy growth also contributes to the transition from laminar to turbulent shear flows. Reddy and Henningson [27] provided a reasonable explanation for the transient energy growth in Poiseuille and Couette flows, namely, the non-modal analysis theory. They also extended their study to pipe flow [28]. Schmid [29,30] built on their work and discussed the impact of the parameters on the results. He also proposed alternative methods to analyze the transient growth mechanism. De Luca et al. [31] researched the energy growth of the initial perturbations in two-dimensional gravitational plane jets. Yecko [32] studied the energy growth in two-phase layers. Li [33] examined the transient growth in channel flow with an electric field. Liu [34] studied the transient growth in the plane Couette flow of a power-law flow. Hu [35] investigated the transient growth in thermocapillary liquid layers. Chaudhary et al. [36] discussed the same phenomenon in a viscoelastic channel flow.

The transient energy growth and parameter discussion in cylindrical jet flows have received little attention in previous studies. Modal analysis has been predominantly used in previous work, which considers the linear superposition of disturbance waves. However, modal analysis cannot explain the transient growth behavior of energy [37]. In some cases, this transient energy growth can have significant impacts on the results. Non-modal analysis, on the other hand, can explain the mechanism of this short-term growth. Previous work has provided two methods to explain this phenomenon. Mathematically, the flow-governing operator is non-orthogonal, leading to coupling between eigenvectors [27]. Physically, the lift-up mechanism is introduced into the physical interpretation, which shows a coupling behavior between different disturbance waves [28]. Based on the above analysis, this paper will analyze the stability of the free cylindrical liquid jet from the perspective of transient energy growth.

In Section 2, we discuss the governing equation and present two methods for solving the transient energy growth problem. Modal analysis is discussed in Section 2.1 and non-modal analysis in Section 2.2. The numerical calculation results are presented in Section 3, where we also discuss the influence of the initial values and parameters and analyze the mechanism of transient energy growth using two approaches: the numerical range and the lift-up mechanism. Finally, we present our conclusions in Section 4.

2. Method

2.1. Governing Equations

The modal presented in Figure 1 is an axisymmetric cylindrical jet with equal diameter. In this model, the jet is assumed to be viscous and incompressible, with a flow radius of R, liquid density of ρ, and uniform basic flow velocity with only axial velocity U₀. To enable more general applications, each parameter is nondimensionalized. The length scale is taken as the radius R of the undisturbed jet, velocity scale as the basic flow velocity U₀, and pressure scale as $\rho U_0^2$. The Reynolds number is defined as $Re=\rho U_{0}R/\mu$ and the Weber number as $We=\sigma /\rho U_0^{2}R$, where σ represents the surface tension coefficient of the interface.

Regardless of circumferential influence, the linearized momentum equation and continuity equation for disturbances in cylindrical coordinates (r, z) can be written as follows:

$$\begin{array}{c}{\displaystyle \frac{\partial u}{\partial z}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \left(rv\right)}{\partial r}}=0\\ {\displaystyle \frac{\partial v}{\partial t}}+U{\displaystyle \frac{\partial v}{\partial z}}=-{\displaystyle \frac{\partial p}{\partial r}}+{\displaystyle \frac{1}{Re}}\left({\nabla }^{2}v-{\displaystyle \frac{v}{r^2}}\right)\\ {\displaystyle \frac{\partial u}{\partial t}}+U{\displaystyle \frac{\partial u}{\partial z}}+v{\displaystyle \frac{\partial U}{\partial r}}=-{\displaystyle \frac{\partial p}{\partial z}}+{\displaystyle \frac{1}{Re}}{\nabla }^{2}u\end{array}$$

(1)

where

$${\nabla }^2={\displaystyle \frac{{\partial }^2}{\partial z^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{{\partial }^2}{\partial r^2}}$$

(2)

$$U=1$$

(3)

Unlike flow in a tube, the cylindrical jet lacks wall support on its outer boundary, leading to disturbance of the jet boundary. Assuming that the surface disturbance is represented by $\eta \left(z,t\right)$, the radius of the cylindrical jet can be expressed as r = 1 + η. The boundary conditions for the equal diameter cylindrical jet are assumed as follows. Linearized motion boundary conditions are imposed at r = 1 + η:

$$r=1+\eta : \mathrm{v}={\displaystyle \frac{\partial \eta }{\partial t}}+U{\displaystyle \frac{\partial \eta }{\partial z}}$$

(4)

After substituting the expression for the mainstream velocity given in Equation (4) into the aforementioned formula, we obtain the following:

$$r=1+\eta : \mathrm{v}={\displaystyle \frac{\partial \eta }{\partial t}}+{\displaystyle \frac{\partial \eta }{\partial z}}$$

(5)

As the viscosity of the gas surrounding the cylindrical jet is neglected, there exists no shear stress on the outer boundary of the jet. Therefore, a tangential boundary condition is imposed at r = 1 + η:

$$r=1+\eta :{\displaystyle \frac{\partial v}{\partial z}}+{\displaystyle \frac{\partial u}{\partial r}}=0$$

(6)

Next, we supplement the boundary conditions at the axis of the cylindrical jet. By exploiting the symmetry, we can infer that the radial velocity is zero at the jet axis. Additionally, the partial derivative of the axial velocity and pressure disturbance in the radial direction also vanishes, which is mathematically expressed as follows:

$$r=0:v={\displaystyle \frac{\partial v}{\partial z}}={\displaystyle \frac{\partial u}{\partial r}}={\displaystyle \frac{\partial p}{\partial r}}=0$$

(7)

The time-dependent behavior of the cylindrical jet is also taken into consideration. The spatial wave number k is a real number, and the various disturbances are expressed in the form of Fourier expansion in space:

$$q(r,z,t)=\widehat{q}(r)e^{ikx-i\omega t}$$

(8)

The aforementioned η (z, t) denotes the displacement of the interface, which is already a perturbation; hence, the original notation is retained.

Linearizing the governing Equation (1) involves expanding various parameters, such as the axial velocity, as the sum of their mean value and disturbance. For instance, the axial velocity can be expressed as the sum of the mean value, U₀, and the disturbance, u. This linearization method is then applied to the governing Equation (1) to derive the equations governing the disturbances.

$$u=U+\widehat{u}$$

(9)

After linearization, the governing equation can be expressed as follows:

$$\left\{\begin{array}{c}ik\widehat{u}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial (r\widehat{v})}{\partial r}}=0\\ {\displaystyle \frac{\partial \widehat{v}}{\partial t}}+ikU\widehat{v}=-{\displaystyle \frac{\partial \widehat{p}}{\partial r}}+{\displaystyle \frac{1}{\mathrm{Re}}}\left({\nabla }^2\widehat{v}+{\displaystyle \frac{\widehat{v}}{r^2}}\right)\\ {\displaystyle \frac{\partial \widehat{u}}{\partial t}}+ikU\widehat{u}=-ik\widehat{p}+{\displaystyle \frac{1}{\mathrm{Re}}}{\nabla }^2\widehat{u}\end{array}\right.$$

(10)

where ${\nabla }^2=-k^2+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{{\partial }^2}{\partial r^2}}.$

In addition, the linearized boundary conditions are specified as follows:

$$\begin{array}{l}r=1+\eta \hspace{1em}\widehat{v}={\displaystyle \frac{\partial \eta }{\partial t}}+ik\eta \\ r=1+\eta \hspace{1em}ik\widehat{v}+{\displaystyle \frac{\partial \widehat{u}}{\partial r}}=0\\ r=0\hspace{1em}\widehat{v}={\displaystyle \frac{\partial \widehat{u}}{\partial r}}={\displaystyle \frac{\partial \widehat{p}}{\partial r}}=0\end{array}$$

(11)

By employing the governing equations and boundary conditions (10) and (11), an accurate description of the disturbance field can be obtained, which enables a modal analysis and non-modal analysis to be conducted.

2.2. Modal Analysis

The modal analysis describes the disturbance field as a superposition of disturbance waves and expresses all disturbance waves in the form of generalized eigenvalue equations. For the jet flow model, the eigenfunction $x=(\widehat{u},\widehat{v},\eta )$ is chosen, and the generalized eigenvalue equations can be written as follows:

$$L_1{\displaystyle \frac{\partial x}{\partial t}}=L_{2}x$$

(12)

As a temporal mode, the solution to the equation can be represented as follows:

$$x(r,t)={\displaystyle \sum _j{a}_j}{e}^{-i{\omega }_{j}t}\tilde{x}(r)$$

(13)

The eigenvalue of the equation represents the complex frequency, with $\omega ={\omega }_r+i{\omega }_i$, where ${\omega }_r$ is the real part, representing the frequency of the wave, and ${\omega }_i$ is the imaginary part, representing the growth rate of the wave. If ${\omega }_i>0$, the wave is considered unstable, indicating that the amplitude of the wave will increase to infinity over time. Conversely, if ${\omega }_i<0$, the wave is stable, suggesting that the perturbation will decay to zero over time. To solve Equation (12), the Chebyshev spectral method can be applied. A grid is set in the r direction, where r ∈ [0, 1]. Moreover, each grid represents a mode, which has its own eigenvalue and eigenvector. In this case, 100 modes are used to solve these equations. A total of 100 modes were used to solve the equations, which is sufficient to capture all modes with slow temporal decay. This allows for analyzing the entire evolution process of jet disturbances. The partial derivative of r can be expressed as follows:

$$D={\displaystyle \frac{\partial }{\partial r}}{D}^2={\displaystyle \frac{{\partial }^2}{\partial r^2}}\hspace{1em}\dots D^n={\displaystyle \frac{{\partial }^n}{\partial r^n}}$$

(14)

Thus, Equation (12) can be expressed as follows:

$$-i\omega L_{1}x=L_{2}x$$

(15)

The matrices L₁ and L₂ are defined as follows:

$$L_1=\left[\begin{array}{ccc}{\displaystyle \frac{D}{r}}+D^2-k^2& 0& 0\\ 0& {\displaystyle \frac{D}{r}}+D^2-k^2& 0\\ 0& 0& 1\end{array}\right]$$

(16)

$$L_2=\left[\begin{array}{ccc}{\mathcal{L}}_{11}& 0& 0\\ 0& {\mathcal{L}}_{22}& 0\\ 0& {\mathcal{L}}_{32}& -ikU\end{array}\right]$$

(17)

where

$$\begin{array}{l}{\mathcal{L}}_{11}=\left\{{\displaystyle \frac{1}{\mathrm{Re}}}\left[D^4-2{\displaystyle \frac{D^3}{r}}+\left({\displaystyle \frac{1}{r^2}}-2k^2\right)D^2-2k^2{\displaystyle \frac{D}{r}}+k^4\right]-ikU\left({\displaystyle \frac{D}{r}}+D^2-k^2\right)\right\}\\ {\mathcal{L}}_{22}=\left\{{\displaystyle \frac{1}{\mathrm{Re}}}\left[D^4-2{\displaystyle \frac{D^3}{r}}+\left({\displaystyle \frac{1}{r^2}}-2k^2\right)D^2-2k^2{\displaystyle \frac{D}{r}}+k^4-{\displaystyle \frac{1}{r^2}}\left({\displaystyle \frac{D}{r}}+D^2-k^2\right)\right]-ikU\left({\displaystyle \frac{D}{r}}+D^2-k^2\right)\right\}\end{array}$$

(18)

These two matrices are of dimension n*n, where n is the number of points in the discrete formulation, the solution of which yields n modes.

Here, L₃₂ denotes a matrix of size n*1 with each element equal to 1.

Firstly, the dispersion equation is calculated to determine the stable mode. The dispersion equation represents the relationship between wave frequency and wave number. The dimensionless dispersion equation of the cylindrical jet can be obtained by solving Equations (10) and (11), and it is expressed as the following formula:

$$D(k,\omega )=-{\displaystyle \frac{k^4-l^4}{R{e}^2}}{\displaystyle \frac{I_0\left(k\right)}{I_1\left(k\right)}}+{\displaystyle \frac{2k^2\left(k^2+l^2\right)}{R{e}^2}}{\displaystyle \frac{I_1^{\mathrm{\prime }}\left(k\right)}{I_1\left(k\right)}}-{\displaystyle \frac{4l_1{k}_1^3}{R{e}^2}}{\displaystyle \frac{I_1^{\mathrm{\prime }}\left(l_1\right)}{I_1\left(l_1\right)}}-kWe\left(1-k^2\right)=0$$

(19)

where I₀ and I₁ are zero-order and first-order Bessel functions, respectively. $\mathrm{\prime }$ means to derive a function, while l is a combined parameter, which can be written as $l^2=k^2+\mathrm{R}\mathrm{e}(ik-i\omega )$.

As shown in Figure 2, the Reynolds number is set to Re = 100 and the Weber number to We = 0.1. It can be observed that, when the wave number k is greater than 1.007, all the modes exhibit negative growth rates. This implies that any disturbance imposed on the flow field will decay over time. Such conditions can be described using modal analysis. It can be clearly observed that there are two regions: an unstable region (k < 1) and a stable region (k > 1). Therefore, one wave number was selected from each region for analysis. First, in unstable region, k = 0.75 is chosen. Figure 2b presents the eigenvalues under the condition of k = 0.75. It can be observed that, under these conditions, there exists an unstable mode with a temporal growth rate greater than 0. As a result, this mode generates disturbances that grow continuously over time, exhibiting exponential growth.

Figure 2c shows the eigenvalues for Re = 100, We = 100, and k = 2. It can be observed that all eigenvalues of the modes are less than 0. Under these conditions, the eigenfunctions of all disturbance modes decay over time. It is important to emphasize that the growth or decay of disturbances cannot be determined solely from the eigenfunctions. This study focuses on the coupling between different modal eigenfunctions, and the effects of this coupling can be revealed through non-modal analysis. Even if all eigenfunctions decay over time, energy growth for the entire flow system may still occur in the short term, which is considered a clear manifestation of transient growth.

With k = 1.5 and Re = 200, the eigenvalues of the equation are plotted in Figure 3a. The frequency of the disturbance wave is shown on the x-axis, and the growth rate is on the y-axis. As shown, the imaginary parts of the top three eigenvalues are −0.002, −0.37, and −0.002, respectively, indicating that the disturbance wave is stable. In addition, the two least stable modes correspond to two capillary waves, one with a large propagation frequency and the other with a small one. These two propagation frequencies are symmetrical about the other propagation frequencies. Moreover, the real part of the eigenvalues approaches 1.52 at k = 1.5, indicating that the phase speed is close to 1, which is consistent with the theoretical phase speed. The above two modes represent two capillary waves propagating in opposite directions, so they are called capillary modes, which are two complex solutions of the dispersion equation of a cylindrical jet. Other eigenvalues characterize the flow field inside the cylindrical jet, which is called the hydrodynamic modes.

Furthermore, when setting k = 2 and Re = 400, the eigenvalues of the equation are depicted in Figure 3b. The three least stable modes correspond to eigenvalues with imaginary parts of −0.02, −0.06, and −0.02, respectively. In this scenario, the real parts of the eigenvalues approach 2.01 as the frequency. The phase speed remains close to 1, indicating the effectiveness of the eigenvalue-solving approach. However, despite its usefulness, modal analysis has limitations. As a linearized method, it cannot accurately solve nonlinear energy problems.

Using the modal analysis method alone, it is not possible to determine whether there is short-term transient growth in the flow field. This is because the method treats each disturbance wave as an independently propagating wave and does not account for coupling between different waves. To address this issue, a non-modal analysis will be introduced in the following section.

2.3. Non-Modal Analysis

The aforementioned modal analysis is conducted based on the generalized eigenvalue equation, in which only the eigenvalues are considered. However, the eigenfunctions, representing velocity disturbances and the interface displacement, can reveal the coupling relationship between various disturbance waves. The energy norm, which quantifies the growth or decay of disturbances, is a crucial indicator. Specifically, the energy norm reflects the change in the disturbance amplitude. Mathematically, the energy norm can be expressed as follows:

$$E=||x{||}^2=E_{\mathrm{kin} }+E_{\sigma }$$

(20)

where E_kin denotes the kinetic energy of the disturbances, and E_σ represents the surface tension energy. These two types of energy can be expressed using the following formulae:

$$\begin{array}{l}{E}_{kin}={\displaystyle \frac{1}{2}}{\displaystyle \int \left(|\widehat{u}{|}^2+|\widehat{v}{|}^2\right)}dr\\ E_{\sigma }={\displaystyle \frac{3k^{2}We}{2\mathrm{Re}}}|\eta {|}^2\end{array}$$

(21)

By utilizing the energy norm, it becomes possible to quantify the amplification of perturbation energy. The growth function, represented by G(t), is defined as the ratio of the energy norm at time t to that at time 0. It can be expressed mathematically as follows:

$$G(t)={\displaystyle \frac{||x(t){||}^2}{||x(0){||}^2}}$$

(22)

The detailed derivation steps were provided by Schmid and Henningson [28]. To compute the amplification, the first K modes should be selected. If K is sufficiently large, the amplification will not change. The first K modes are denoted by the following:

$$x(r,t)\approx {\displaystyle \sum _{j=1}^K{a}_j}{e}^{-i{\omega }_{j}t}{\tilde{x}}_j(r)$$

(23)

Here, the coefficient of the eigenvector is denoted by a_j.

As the transient growth is dependent on the non-orthogonality of each eigenvector, we utilize the inner product to quantify this parameter:

$$A=\left(x_i,x_j\right)={\displaystyle \int \left({\widehat{u}}_i^{\ast }{\widehat{u}}_j+{\widehat{v}}_i^{\ast }{\widehat{v}}_j+{\eta }_i^{\ast }{\eta }_j\right)}dr$$

(24)

Here, A can be decomposed into the form $A=F^{\ast }F$, where $F^{\ast }$ is the conjugate of F. Therefore, the amplification can be expressed as follows:

$$\begin{array}{l}G(t)={\displaystyle \frac{||x(t){||}^2}{||x(0){||}^2}}={‖F\exp \left(-i{\Lambda }_{K}t\right)F^{-1}‖}_2^2\\ G(\max )={\max }_{t⩾0}G(t)\end{array}$$

(25)

The 2-norm of matrix x is denoted by ‖x‖₂, and can be calculated using the singular value decomposition (SVD) method. Λ_K is a matrix with diagonal elements composed of the eigenvalues of the K least stable modes. By computing the growth function G(t) at different time instances, we can determine the optimal transient energy growth. If G(t) is greater than 1, it indicates the presence of transient energy growth in the jet flow field.

3. Results

3.1. Optimal Growth

The parameters used in this study are k = 1.5, Re = 200, We = 10, and K = 100 for a jet flow field. As evident from the spectrum presented earlier, all the eigenvalues are less than 0 under these conditions, indicating that the Reynolds number is less than the critical Re. As per the conclusions drawn from the modal analysis, there is no transient growth, and a disturbance will decay to zero with time. However, upon utilizing the non-modal analysis, transient growth can be clearly observed in Figure 4a.

As observed, even though the Reynolds number and the Weber number are relatively small, a transient energy growth can still be observed in the initial few seconds. The peak growth is observed at t = 4.54, with a maximum value of G(max) = 9.22. However, this growth eventually decays to zero over time. Furthermore, growth rates that are less than −100 are ignored, as these modes are considered stable. In this study, 101 modes were utilized to obtain the results. Increasing the number of modes K may yield more options for analysis.

By increasing the time range to [0, t_max], we can observe additional energy growth with an amplification greater than 1. This is demonstrated in Figure 4b, where we set t_max = 30.

As observed, the optimal transient growth occurs at t = 4.54, and, subsequently, two additional energy growth events are found with G(t) > 1, as shown in Figure 4b. The first energy growth event is observed at t = 13.64 with a growth factor of G(t) = 7.23. This phenomenon shows that, in the jet flow field, the transient energy growth does not necessarily occur only once, and may even occur multiple times under some conditions. But the amplitude of each transient energy growth will decrease with time. The subsequent transient increase with a smaller amplification is actually caused by considering the existence of surface energy. The oscillation of surface energy and the reduction in kinetic energy actually produce a competition mechanism, and this competition mechanism leads to the oscillation of the entire energy amplification.

In addition, similar phenomena can be observed when the flow parameters are changed. By setting k = 2, Re = 400, and We = 10 and choosing the time range [0, 10], the opposite conclusion to that of the modal analysis can be observed. The results are presented in Figure 5, where the amplification reaches G(t) = 12.44 at t = 2.32.

Therefore, when addressing the problem of energy growth in the short term, non-modal analysis enables us to capture the instantaneous energy evolution of the disturbance at any given moment, while modal analysis only provides information on the overall trend of the energy change over a longer period. Given the nonlinearity inherent in the growth of the disturbance energy, it is crucial to consider the interplay between the disturbance waves, thus making the non-modal analysis more accurate than modal analysis. Furthermore, the energy increase in this regime cannot be overlooked in some instances and may even lead to the premature breakage of the jet. The impact of each dimensionless parameter on the transient growth results will be discussed in the following section.

3.2. Optimal Initial Condition

The optimal initial condition is defined by the following formula [38]:

$$G^{opt}={\max }_{k}G(\max )$$

(26)

As depicted in Figure 2, there exists a critical wave number such that, for values of k exceeding this threshold, the temporal growth rate of all disturbance waves is less than 0. Therefore, it becomes imperative to investigate the optimal initial condition for maximum amplification growth when the wave number falls into two distinct regimes.

First, when the wave number k exceeds the critical number, we set Re = 500 and We = 0.001. Under this condition, the critical number is slightly less than 1.01. To analyze short and long waves, we select wave numbers k = 2, 2.5, 3, and 5, and the results are presented in Figure 6. It is observed that, when k = 2, the amplification is less than 10 due to its proximity to the critical wave number. However, when the initial condition has a larger wave number, the amplification increases. Additionally, the time t corresponding to G(max) advances.

Conversely, as the wave number increases to a specific value, the maximum amplification decreases. At a constant Reynolds and Weber number, we choose wave numbers k = 60, 65, 70, and 75 within the time range of 0 to 0.2, and the results are presented in Figure 7.

The findings indicate that there exists a negative correlation between the maximum amplification and wave number when the latter is large. Furthermore, the time t corresponding to G(max) advances with increasing wave number. To calculate the optimal transient growth, we select wave numbers between 20 to 80 and set the Reynolds number Re = 500 and the Weber number We = 0.001. The results are presented in Figure 8.

It can be observed from Figure 8 that, for a given Reynolds and Weber number, the maximum amplification first increases and then decreases as the wave number increases. Furthermore, there exists an optimal initial disturbance under the condition of We = 0.001 and Re = 500. When the initial wave number is k = 51.58, the maximum value of transient energy growth G(max) = 42.50.

The wave number k affects the disturbance energy from two perspectives. Firstly, increasing k reduces the overall growth rate and accelerates the decay of the disturbance. Secondly, based on Equation (23), the wave number k is positively correlated with the surface tension energy. However, the results indicate that, within the range of wave numbers considered, the surface energy initially dominates, leading to an increase in maximum amplification as the wave number increases. As the wave number further increases, kinetic energy gradually takes over, resulting in a smaller growth rate and faster attenuation of kinetic energy, thereby causing a continuous decrease in the maximum amplification.

Moreover, when the wave number k is less than the critical wave number, modal analysis shows that one of the growth rates of the disturbance wave can be greater than 0, as illustrated in Figure 9.

Under such conditions, the mode is unstable and the disturbance will eventually grow infinitely. Therefore, discussing the initial conditions for optimal growth is not meaningful. However, transient growth still exists during the initial phase, as illustrated in Figure 10. For this analysis, we choose k = 0.5, Re = 200, and We = 1. Considering the capillary modes under the influence of surface waves [38], we can study the impact of energy magnification related to the surface wave energy and surface energy, which dominates the R–P instability mechanism.

In classical theory, when a mode is unstable, the disturbance will grow exponentially. However, in this case, it is observed that the disturbance grows faster than exponential growth during the initial period. The growth during this period not only has linear exponential growth but also transient growth. As shown in the figure, the growth rate at the initial moment is significantly greater than the exponential growth rate. Thus, the occurrence of transient growth can explain why the energy of the cylindrical jet system grows more rapidly.

3.3. The Influence of Reynolds Number

Here, we set k = 1.5, We = 0.1, Re = 100, 500, 1000, and 2000 and the results are shown in Figure 11a:

From the results presented, it can be observed that there is a direct proportionality between the amplification for G(max) and the Reynolds number. When Re varies from 100 to 2000, the amplification shows an increasing trend. To further investigate this phenomenon, we set k = 2 and We = 0.1, and choose Reynolds numbers of 100, 500, 1000, and 2000. The similar trend is observed, as depicted in Figure 11b. The results indicate that the transient energy growth is positively correlated with the Reynolds number. Moreover, it is observed that the time t corresponding to G(max) is delayed as the Reynolds number increases.

The mechanism underlying this phenomenon can be explained through the energy formula. As the Reynolds number increases, the disturbance kinetic energy also increases, which leads to an amplification of the initial transient energy growth. This indicates that, as the Reynolds number changes, the amplification is influenced by both the disturbance kinetic energy and the surface tension energy. Among these two factors, the impact of the disturbance kinetic energy is greater. As a result, the maximum amplification, G(max), demonstrates a trend of increasing as the Reynolds number increases.

3.4. The Influence of Weber Number

Here, we consider the case where k = 1.5 and Re = 200, and investigate the effect of the Weber number on the system. Specifically, we choose several values of the Weber number, namely, We = 0.05, 0.1, 0.2, and 1. The results are presented in Figure 12a. Similarly, when we set k = 2 and Re = 200, the same trend can be observed for different values of the Weber number, as shown in Figure 12b. It is important to emphasize that our study revealed that, under small wave number conditions, the temporal growth rates of each mode are similar, allowing for data analysis over a longer time scale. In contrast, under large wave number conditions, significant differences in the temporal growth rates between modes arise, and transient growth phenomena occur on shorter time scales. Therefore, in previous studies [27,29], researchers often analyzed disturbances with small wave numbers to derive subsequent research conclusions.

It is evident from the results presented in Figure 12a,b that the maximum amplification G(max) decreases continuously as the Weber number increases, and the corresponding time is gradually delayed. For the case of k = 1.5, it can be observed that, when We = 0.05, the maximum growth reaches G(max) = 3.31 at t = 6.06. In contrast, when We = 1, the maximum growth is G(max) = 4.48 at t = 1.52. Similarly, for the case of k = 2, it can be seen that, when We = 0.05, the maximum growth is G(max) = 6.05 at t = 3.18, whereas, when We = 1, the maximum growth is G(max) = 8.12 at t = 0.76. It is noteworthy that, for both wave number conditions, the maximum amplification increases with the enhancement of the Weber number, and the corresponding time decreases with the enhancement of the Weber number.

Based on the definition of the Weber number and Equation (20), it can be observed that, when the Weber number is low, the surface tension energy is relatively small. However, when the Weber number is sufficiently large, the surface tension energy becomes significant, gradually becoming a more dominant factor in determining the result. At the boundary of the jet, the amplification G(max) is mainly governed by the surface tension energy. Moreover, as the Weber number increases, the surface tension energy increases at a faster rate. Therefore, the corresponding time t for G(max) will occur earlier as the Weber number increases.

3.5. Numerical Range

As previously stated, the modal analysis method involves calculating the eigenvalues of the governing operator of the generalized eigenvalue equation and observing the evolution of the disturbance wave over time by calculating the imaginary part of the eigenvalue. However, this method is limited to predicting the long-term behavior of the disturbance wave.

In this study, a straightforward approach is employed to estimate the short-term development of the disturbance wave energy growth [39]. Specifically, the amplification G(t) is defined as the energy norm amplification at time t. To evaluate this, a Taylor series expansion of the matrix exponential term of the energy norm is conducted at t → 0⁺. This method enables us to obtain a concise expression for the disturbance energy growth in the early stage of development:

$$\exp \left(-i{\Lambda }_{K}t\right)\approx 1+t\left(-i{\Lambda }_K\right)+\cdots$$

(27)

Introducing the following definition:

$$L=-i{\Lambda }_K$$

(28)

Therefore, the energy growth rate in the short term can be calculated as follows:

$$\begin{array}{l}{\left.{\displaystyle \frac{dG(t)}{dt}}\right|}_{t\to 0^{+}}={\left.{\max }_{x(0)}{\displaystyle \frac{1}{||x(0){||}^2}}{\displaystyle \frac{d}{dt}}||(1+tL)x(0){||}^2\right|}_{t\to 0^{+}}\\ ={\left.{\max }_{x(0)}{\displaystyle \frac{d}{dt}}{\displaystyle \frac{\langle (1+tL)x(0),(1+tL)x(0)\rangle }{\langle x(0),x(0)\rangle }}\right|}_{t\to 0^{+}}\\ ={\left.{\max }_{x(0)}{\displaystyle \frac{〈x(0),\left(L+L^H\right)x(0)〉}{\langle x(0),x(0)\rangle }}\right|}_{t\to 0^{+}}\\ ={\omega }_{\max }\left(L+L^H\right)\end{array}$$

(29)

where x (0) denotes the initial eigenvector of the matrix L + L^H at t = 0.

In accordance with this formula, the short-term energy growth can be expressed by the eigenvalue of the matrix L + L^H instead of the matrix L. This eigenvalue is commonly known as the Rayleigh quotient of the matrix L + L^H. To calculate the maximum Rayleigh quotient, the principal eigenvector x (0) is utilized. The resulting eigenvalue of L + L^H represents a slope of the eigenvalue of L, which is demonstrated in Figure 13b. In this study, we consider k = 5, We = 10, and Re = 200, and display the eigenvalue of L in Figure 13a.

We can define the energy growth as transient growth in the short-term and long-term growth. The short-term growth can be determined by the eigenvalue of L + L^H, while the long-term growth can be determined by the eigenvalue of L. The eigenvalue of L + L^H, depicted as the blue slope in Figure 13b, represents the numerical range of L. As time progresses, this envelope gradually converges to the eigenvalue of L, which is the black point. As illustrated in Figure 13a, all the imaginary parts of the eigenvalue are negative. However, the blue slope in Figure 13b extends into the upper half plane, indicating that there exist some eigenvalues with positive imaginary parts in the short term. Over time, the envelope gradually shrinks into the lower half plane. This finding provides an explanation for the observed transient growth in energy at the initial time. At t → 0⁺, the dynamics of the jet flow system are governed by the eigenvalues of L + L^H, some of which are greater than 0. Thus, the energy of the disturbance wave can increase with time in the short term. However, as time progresses, the system is eventually governed by the eigenvalues of L, which are all less than 0 and decay over time in the long term.

4. Conclusions

This paper examines the transient energy growth in cylindrical jet flows with a uniform speed profile. In such flows, there exists a short-term transient growth that can cause the free jet to become unstable or even rupture prematurely. The transient growth is maximized for specific initial conditions. Moreover, the magnitude of the transient growth is affected by several parameters, including the Reynolds number, the Weber number, and the initial conditions.

The R–P instability mechanism highlights the dominance of surface energy over kinetic energy. This phenomenon occurs under low mainstream speeds, where the fluctuations in energy are largely governed by changes in kinetic energy and surface energy oscillations. In this context, the inclusion of surface energy explains the occurrence of multiple transient growths with smaller amplifications after the maximum growth, instead of a single extreme value.

Under specific Reynolds and Weber numbers, there is a particular wave number that results in the maximum amplification. This phenomenon is caused by the competition between surface energy and internal kinetic energy. Moreover, the time corresponding to the optimal growth becomes earlier as the wave number increases. As the Reynolds number increases, the maximum transient energy growth also gradually increases. Similarly, an increase in the Weber number leads to an increase in the maximum transient growth, indicating a positive correlation.

There are two explanations for this phenomenon. Firstly, the numerical range of the governing operator of the flow will extend to the upper half plane of the eigenvalue spectrum, which means that the disturbance growth rate in the short term is positive. Secondly, the existence of the free interface of the jet flow leads to velocity disturbance and boundary displacement affecting each other, resulting in a transient energy increase.