 Research
 Open access
 Published:
Adaptive waveparticle decomposition in UGKWP method for highspeed flow simulations
Advances in Aerodynamics volumeÂ 5, ArticleÂ number:Â 25 (2023)
Abstract
With waveparticle decomposition, a unified gaskinetic waveparticle (UGKWP) method has been developed for multiscale flow simulations. With the variation of theÂ cell Knudsen number, the UGKWP method captures the transport process in all flow regimes without theÂ kinetic solverâ€™s constraint on the numerical mesh size and time step being determined by the kinetic particle mean free path and particle collision time. In the current UGKWP method, the cell Knudsen number, which is defined as the ratio of particle collision time to numerical time step, is used to distribute the components in the waveparticle decomposition. The adaptation of particles in theÂ UGKWPÂ method is mainly for the capturing of the nonequilibrium transport. In this aspect, the cell Knudsen number alone is not enough to identify the nonequilibrium state. For example, in the equilibrium flow regime with a Maxwellian distribution function, even at a large cell Knudsen number, the flow evolution can be still modelled by the NavierStokes solver. More specifically, in the near space environment both the hypersonic flow around a space vehicle and the plume flow from a satellite nozzle will encounter a far field rarefied equilibrium flow in a large computational domain. In the background dilute equilibrium region, the large particle collision time and a uniform small numerical time step can result in a large local cell Knudsen number and make theÂ UGKWPÂ method track a huge number of particles for the far field background flow in the original approach. But, in this region the analytical wave representation can be legitimately used in theÂ UGKWPÂ method to capture the nearly equilibrium flow evolution. Therefore, to further improve the efficiency of theÂ UGKWPÂ method for multiscale flow simulations, an adaptive UGKWP (AUGKWP) method is developed with the introduction of an additional local flow variable gradientdependent Knudsen number. As a result, the waveparticle decomposition in theÂ UGKWPÂ method is determined by both theÂ cell and gradient Knudsen numbers, and the use of particles in theÂ UGKWPÂ method is solely to capture the nonequilibrium flow transport. The current AUGKWPÂ method becomes much more efficient than the previous one with the cell Knudsen number only in the determination of waveparticle composition. Many numerical tests, including Sod shock tube, normal shock structure, hypersonic flow around cylinder, flow around reentry capsule, and an unsteady nozzle plume flow, have been conducted to validate the accuracy and efficiency of the AUGKWP method. Compared with the original UGKWP method, the AUGKWP method achieves the same accuracy, but has advantages in memory reduction and computational efficiency in the simulation for flows with the coexisting of multiple regimes.
1 Introduction
Around a highspeed flying vehicle with complex irregular largescale configurations in nearspace flight surrounding, the highly compressed gas nearby the leading edge and the strong expansion wave around the leeward zone cover multiflow zone mixed flows with huge pressure and density differences due to the influence of viscosity and adverse pressure gradients at the same flying altitude [1]. In the control system of a moving satellite, the flow expansion inside a nozzle can undergo a rapid and unstable transition from continuum to freemolecule flow as the local Knudsen number varies by an order of magnitude of ten. This transition occurs as theÂ highpressure gas inside the nozzle expands into the background vacuum. Multiscale flows involve a large variation of Knudsen number and significant changes of the degrees of freedom in the description of flow physics. In aerospace applications, an accurate and efficient multiscale method with the capability of simulating both continuum and rarefied flows is of great importance [2].
The Boltzmann equation is the fundamental governing equation for rarefied gas dynamics. Theoretically, it can capture multiscale flow physics in all Knudsen regimes, with the enforcement to the modeling scales of the Boltzmann equation, such as the particle mean free path and mean collision time. For aÂ nonequilibrium flow, there are mainly two kinds of numerical methods to solve the Boltzmann equation, i.e., the stochastic particle method and the deterministic discrete velocity method. The stochastic methods employ discrete particles to simulate the statistical behavior of molecular gas dynamics [1, 3,4,5,6,7,8,9,10,11]. This kind of Lagrangiantype scheme achieves high computational efficiency and robustness in rarefied flow simulation, especially for hypersonic flow. However, it suffers from statistical noise in the lowspeed flow simulation due to its intrinsic stochastic nature. Meanwhile, in the near continuum flow regime, the particle method becomes very expensive due to the requirement of small cell size and time step in the computational space and the tracking of a large number of particles with intensive collisions. The deterministic approaches use discrete particle velocity space to solve the kinetic equation and naturally obtain accurate solutions without statistical noise [12,13,14,15,16,17,18,19,20,21,22,23,24,25]. At the same time, the deterministic method can achieve high efficiency using numerical acceleration techniques, such as implicit algorithms [14, 13, 26,27,28,29], memory reduction techniques [24], adaptive refinement methods [30], and fast evaluation of the Boltzmann collision term [31, 32]. Asymptotic preserving (AP) schemes [33, 34] and unified preserving (UP) schemes [35] can be developed to release the stiffness of the collision term at the small Knudsen number case. However, for most AP schemes only the Euler solution in the hydrodynamic limit is recovered, where the NS limit can be obtained in the AP schemes. Additionally, for hypersonic and rarefied flows, the deterministic methods have to use a gigantic number of grid points in the particle velocity space to cover a large variation of particle velocity and resolve the nonequilibrium distribution. The threedimensional hypersonic flow calculation can be hardly conducted due toÂ the huge memory consumption and computational cost. Moreover, for both stochastic and deterministic methods, with the operatorsplitting treatment of particle free transport and instant collision, a numerical dissipation proportional to the time step is usually unavoidable. Therefore, the mesh size and the time step in both theÂ traditional particle method, such as DSMC, and the deterministic DVM method, are required to be less than the particle mean free path and collision time, to avoid the physical dissipation being overwhelmingly taken over by the numerical ones, especially in the continuum flow regime, such as the laminar boundary layer computation at aÂ high Reynolds number [36]. In order to remove the constraints on the mesh size and time step in the continuum flow regime, the unified gaskinetic scheme (UGKS) and discrete UGKS (DUGKS) with the coupled particle transport and collision in the flux evaluation have been constructed successfully [18, 29, 37]. At the same time, the multiscale particle methods have been constructed as well [38, 39].
Combining the advantages of the deterministic and the stochastic methods, a unified gaskinetic waveparticle (UGKWP) method [40, 41] has been proposed under the UGKS framework [2, 42], as well as simplified versions [43, 44]. The method has included molecular rotation and vibration [45, 46], and has beenÂ extended to other multiscale transports, such as radiation, plasma, and multiphase flow [47,48,49]. Taking advantage of the evolution solution of theÂ kinetic model equation [50] in the scheme construction, the UGKWP method can capture the flow physics in all flow regimes and release the restrictions on the mesh size and time step which are previously imposed on the kinetic solvers. In theÂ rarefied flow region with a large cell Knudsen number, the UGKWP method becomes a particle method for the capturing of peculiar gas distribution function. In the continuum region with a small cell Knudsen number, the UGKWP method gets back to a hydrodynamic flow solver, where the gaskinetic scheme (GKS) for the NavierStokes solution [51] is fully recovered. Different from domain decomposition methods with buffer zones, the UGKWP method employs waveparticle decomposition in each cell with a unified treatment in the whole computational domain. The essential criterion used in the UGKWP method to identify different flow regimes is according to the cell Knudsen number \({Kn}_c = \tau / {\Delta t}\), where \(\tau\) is the local particle collision time and \(\Delta t\) is the numerical time step. Naturally the cell Knudsen number controls the weights in the waveparticle decomposition. However, besides identifying the real nonequilibrium flow region, the large cell Knudsen number also picks up a dilute equilibrium state as the rarefied regime and uses particles to simulate the background equilibrium flow evolution. For example, the far field of hypersonic flow around a space vehicle is usually in an equilibrium state with a Maxwellian distribution, and the cell Knudsen number in the background equilibrium state is large. The large cell KnudsenÂ number will increase the weight of particles while in reality the analytical wave can be faithfully used in this region. In this paper, besides the cell Knudsen number, a new parameter for identifying the local nonequilibrium state will be introduced and added to the previous criteria for waveparticle decomposition. As a result, an adaptive unified gaskinetic waveparticle method (AUGKWP) will be constructed with a clear identification of theÂ nonequilibrium region. The new method will avoid using particles to the background dilute equilibrium region and further improve the computational efficiency of theÂ UGKWPÂ method.
The paper is organized as follows. Since the UGKWPÂ method is an enhanced version of the solely particlebased unified gaskinetic particle (UGKP) method by employing both wave and particle compositions, the UGKP method will be introduced first in Section 2. The UGKWP method will be discussed in Section 3. Then the adaptive UGKWPÂ method by mainly concentrating particle distribution to the nonequilibrium flow region will be presented in Section 4. Numerical validation of the current method will be carried out in Section 5 and a conclusion will be drawn in Section 6.
2 Unified gaskinetic particle method
2.1 General framework
The unified gaskinetic particle (UGKP) method is a particle implementation of the UGKS under the finite volume framework, where the discrete particles are employed to describe the nonequilibrium gas distribution function, and the evolution of particles recovers the multiscale nature in different flow regimes. The kinetic equation with BGK relaxation model is
where the equilibrium state g is the Maxwellian distribution function,
where d is the degrees of freedom, and \(\lambda\) is related to the temperature T by \(\lambda = m_0/2k_BT\). Here, \(m_0\) and \(k_B\) are the molecular mass and Boltzmann constant, respectively. \(\varvec{c} = \varvec{u}  \varvec{U}\) denotes the peculiar velocity. Along the characteristic line, the integral solution of the kinetic model equation gives
where \(f_0(\varvec{r})\) is the initial distribution function at the beginning of each step \(t_n\), and \(g(\varvec{r}, t)\) is the equilibrium state distributed in space and time around \(\varvec{r}\) and t. The integral solution describes an evolution process from theÂ nonequilibriumÂ state to theÂ equilibrium state through particle collisions.
In the UGKS, with the expansion of theÂ initial distribution function and theÂ equilibrium state
the secondorder accurate flux for macroscopic flow variables across cell interface ij can be constructed from the integral solution,
where \({\varvec{n}_{ij}}\) is the normal vector of the cell interface, and
\(\varvec{F}^{fr}_{ij}\) and \(\varvec{F}^{eq}_{ij}\) are the macroscopic fluxes from the free transport and collision processes, respectively. The integrated time coefficients are
The UGKS updates both the gas distribution function and macroscopic flow variables under a finite volume framework. In the UGKP method, the particle will be used to follow the evolution of theÂ gas distribution function directly and keep the finite volume version for the updates of macroscopic flow variables. On the microscopic scale, the particle evolution follows the evolution solution in Eq. (2), where the particle free transport and collision will be taken into account. On the macroscopic scale, the fluxes across the cell interface for the updates of macroscopic flow variables inside each control volume are evaluated by Eq. (4).
Denote a simulation particle as \(P_k(m_k,\varvec{r}_k,\varvec{u}_k)\), which represents a package of real gas molecules at location \(\varvec{r}_k\) with particle mass \(m_k\) and microscopic velocity \(\varvec{u}_k\). According to the evolution solution, the cumulative distribution function of particleâ€™s collision is
then the free transport time of a particle within one time step \(\Delta t\) will be
where \(\eta\) is a random number uniformly distributed in (0,Â 1). In a numerical time step from \(t^{n}\) to \(t^{n+1}\), according to the free transport time \(t_f\), the simulation particles can be categorized into collisionless particles (\(t_f = \Delta t\)) and collisional particles (\(t_f < \Delta t\)).
In the free transport process, i.e., \(t < t_f\), no collisions will happen, and the particles move freely and carry the initial information. The trajectory of particle \(P_k\) could be fully tracked by
During the free transport process, the effective net flux across interfaces of cell i can be evaluated by
where \(\varvec{\phi }_{k} = (m_k, m_k \varvec{u}_k, \frac{1}{2} m_k \varvec{u}_k^2)^T\). The free transport flux \(\varvec{F}_{ij}^{fr}\) in Eq. (4) has been recovered by the particlesâ€™ movement.
In the free transport process, the particle during the time interval \((0, t_f)\) is fully tracked. The collisionless particles with \(t_f = \Delta t\) are kept at the end of the time step. The collisional particles with \(t_f < \Delta t\) would encounter collisions at \(t_f\) and they are only tracked up to this moment. Then, all collisional particles are removed, but their accumulated mass, momentum, and energy inside each cell can be still updated through the evolution of macroscopic variables. These collisional particles can be resampled from the updated macroscopic variables at the beginning of theÂ next time step from theÂ equilibrium state if needed.
The equilibrium flux \(\varvec{F}_{ij}^{eq}\) in Eq. (4) contains three terms, i.e., g, \(\partial _{\varvec{r}} g\) and \(\partial _t g\), which are only related to the equilibrium states and can be fully determined by the macroscopic flow variables. Once the Maxwellian distribution and its derivatives around the cell interface are determined, the equilibrium flux \(\varvec{F}^{eq}_{ij}\) can be obtained by
The macroscopic variables for the determination of theÂ equilibrium state \(g_0\) at cell interface ij are coming from the colliding particles from both sides of the cell interface,
where \(\bar{u}_{ij}=\varvec{u}\cdot \varvec{n}_{ij}\) and H[x] is the Heaviside function. The gradient of the equilibrium state is obtained from the gradient of macroscopic flow variables \({\partial \varvec{W}_{ij}}/{\partial {\varvec{r}}}\). In this study, the spatial reconstruction of macroscopic flow variables is carried out by the leastsquare method with Venkatakrishnan limiter [52]. As to the temporal gradient, the compatibility condition on Eq. (1)
is employed to give
Correspondingly, the temporal gradient of theÂ equilibrium state \(\partial _t g\) can be evaluated from the above \({\partial \varvec{W}_{ij}}/{\partial t}\). With \(g_0\), \(\partial _{\varvec{r}} g\) and \(\partial _t g\), the equilibrium flux \(\varvec{F}_{ij}^{eq}\) can be fully determined.
2.2 Updates of macroscopic variables and discrete particles
Under the finite volume framework, according to the conservation law, the updates of macroscopic variables can be written as
where \({\varvec{W}}^{fr}_{i}\) is the net free streaming flow of cell i calculated by particle tracking in the free transport process in Eq.Â (7), andÂ the equilibrium flux \({\varvec{F}}^{eq}_{ij}\) is evaluated from macroscopic flow variables and their gradients in Eq. (8).
Substituting Eq.Â (3) into the integral solution Eq.Â (2) of theÂ kinetic model equation, the time evolution of theÂ distribution function along the characteristic line is
where
It indicates that the collisional particles will follow the nearequilibrium state \(g(\varvec{r}^\prime ,t^\prime )\) after collision within the time step \(t_f < \Delta t\). With the updated macroscopic flow variables, these untracked collisional particles within the time \(t \in (t_f, \Delta t)\) can be resampled from the hydroparticle macroscopic quantities,
where \(\varvec{W}_i^{p, n+1}\) is from the collisionless particles remaining in cell i. With the macroscopic quantities and the form of theÂ equilibrium state g, the corresponding particles can be generated. Details of sampling from a given distribution function are provided in [41].
The free transport and collision processes for both microscopic discrete particles and macroscopic flow variables have been described above. Here, we give a summary of the procedures of the UGKP method. Following the illustration in [41], the algorithm of theÂ UGKP method for diatomic gases with molecular translation, rotation and vibration can be summarized as follows.

Step 1
For the initialization, sample particles from the given initial conditions as shown in Fig.Â 1(a).

Step 2
Generate the free transport time \(t_f\) for each particle by Eq.Â (5), and classify the particles into collisionless particles (white circles in Fig.Â 1(b)) and collisional ones (solid circles in Fig.Â 1(b)). Stream the particles for theÂ free transport time by Eq.Â (6), and evaluate the net free streaming flow \(\varvec{W}^{fr}_i\) by Eq.Â (7).

Step 3
Reconstruct macroscopic flow variables and compute the equilibrium flux \(\varvec{F}^{eq}_{ij}\) by Eq.Â (8).

Step 4
Update the macroscopic flow variables \(\varvec{W}_i\) by Eq.Â (9). Obtain the updated hydroparticle macroscopic quantities of collisional particles \(\varvec{W}^h_i\) by extracting the macroquantities of collisionless particles \(\varvec{W}^p_i\) from the total flow variables \(\varvec{W}_i\) in Eq.Â (10) as shown in Fig.Â 1(c).

Step 5
Delete the collisional particles at \(t_f\) and resample these particles from the updated hydroparticle macroscopic variables \(\varvec{W}^h_i\) as shown in Fig.Â 1(d), which becomes the initial state in Fig.Â 1(a) at the beginning of theÂ next time step.

Step 6
Go to Step 2. Continue time step evolution until the finishing time.
3 Unified gaskinetic waveparticle method
In the UGKP method, based on the updated hydroparticle macroscopic variables \(\varvec{W}^h_i\) of collisional particles, these particles will be resampled from theÂ equilibrium state at the beginning of theÂ next time step. However, some of these resampled particles will get collision in the next time step and get eliminated again. Therefore, in the unified gaskinetic waveparticle (UGKWP) method, only free transport particles in the next time step will be resampled from \(\varvec{W}^h_i\). In the continuum regime at aÂ very small Knudsen number, it is possible that no free particles will get resampled.
The collisionless particles with \(t_f = \Delta t\) will be sampled from \(\varvec{W}^h_i\). According to the integral solution, the collisionless particles will take a fraction of \(\varvec{W}^h_i\) by the amount,
As shown in Fig.Â 2, there is no need to sample these collisional particles from the hydrodynamic portion with macroscopic variables \((\varvec{W}^h_i {\varvec{W}}^{hp}_i)\). The free transport flux from these unsampled collisional particles can be evaluated analytically,
where
Then, the update of macroscopic flow variables in the UGKWP method becomes
The algorithm of the UGKWP method for diatomic gases can be summarized as follows.

Step 1
For the initialization, sample collisionless particles from \(\varvec{W}_i^{hp}\) with \(t_f = \Delta t\) as shown in Fig.Â 2(a). For the first step, \(\varvec{W}^h_i = \varvec{W}_i^{n = 0}\).

Step 2
Generate the free transport time \(t_f\) by Eq.Â (5) for the remaining particles from theÂ previous step evolution with total amount \(\varvec{W}_i^p\), and classify the particles into collisionless particles (white circles in Fig.Â 2(b)) and collisional ones (solid circles in Fig.Â 2(b)). Stream the particles for theÂ free transport time by Eq.Â (6), and evaluate the net free streaming flow \(\varvec{W}^{fr}_i\) by Eq.Â (7).

Step 3
Reconstruct macroscopic flow variables and compute the free transport flux of collisional particles \(\varvec{F}_{ij}^{fr,h}\) by Eq.Â (12) and the equilibrium flux \(\varvec{F}^{eq}_{ij}\) by Eq.Â (8).

Step 4
Update the macroscopic flow variables \(\varvec{W}_i\) by Eq.Â (13). Obtain the updated macroscopic quantities for collisional particles \(\varvec{W}^h_i\) by extracting the macroquantities of collisionless particles \(\varvec{W}^p_i\) from the total flow variables \(\varvec{W}_i\) in Eq.Â (10) as shown in Fig.Â 2(c).

Step 5
Delete the collisional particles at \(t_f\) (\(t_f < \Delta t\)). Resample the collisionless particles from \(\varvec{W}^{hp}_i\) with \(t_f = \Delta t\) at the beginning of theÂ next time step, as shown in Fig.Â 2(d).

Step 6
Go to Step 2. Continue time evolution until the output time.
The UGKP methoï»¿d uses particles to represent the gas distribution function. However, the UGKWP method adopts a hybrid formulation of wave and particles to recover the gas distribution function. Within the time step, the evolution of wave part for the collisional particles (\(t_f < \Delta t\) ) can be described analytically by the time accurate solution of macroscopic flow variables without sampling these particles explicitly. The evolution of the remaining particles will track the nonequilibrium effect through particle free transport. In the rarefied flow regime, the UGKWP method is dominated by particle evolution, which results in a particle method. While in the continuum regime, the UGKWP method is mainly about the evolution of macroscopic variables, and the scheme becomes a hydrodynamic NS solver, such as the socalled gaskinetic scheme (GKS) [51]. Therefore, the UGKWP method achieves much better computational efficiency and lower memory consumption than the pure particle method UGKP.
However, the weights of wave and particles in the current UGKWPÂ method are controlled by the cell Knudsen number \(\tau /\Delta t\). The total mass fraction of particles is proportional to \(e^{\Delta t/\tau } \varvec{W}\). There are still weaknesses in the above formulation. For example, even in the continuum flow regime, if a small time step \(\Delta t\) is used, the particles will emerge automatically in the flow evolution. At the same time, for a dilute background equilibrium distribution in theÂ near space environment with a large particle collision time \(\tau\), the UGKWPÂ method will use the particles to capture the background equilibrium flow evolution. Therefore, besides the cell Knudsen number, in order to use particles to really capture the evolution of theÂ nonequilibrium state, another parameter has to be designed as well in the determination of the distributions between wave and particle in theÂ UGKWPÂ method.
4 Adaptive unified gaskinetic waveparticle method
For the UGKWP method, the evaluation of flow regimes depends on the cell Knudsen number. However, this criterion cannot capture the real nonequilibrium regime, especially for the rarefied undisturbed equilibrium flow and the flow simulation with a very small numerical time step for high resolution. Even in the equilibrium regime, the large particle collision time and the small time step can give a large cell Knudsen number for particle generation. Theoretically, the analytical wave can be used in those equilibrium regimes. Besides the cell Knudsen number, the adaptive unified gaskinetic waveparticle (AUGKWP) method will be developed by introducing another parameter to identify the real nonequilibrium region for the generation of particles. This parameter is a gradientlength related local Knudsen number \({Kn}_{Gll}\). In other words, the analytical wave will take effect as well when \({Kn}_{Gll}\) is small. Therefore, the portion from the macroscopic flow variables to sample particles becomes
where
and the gradientlength local Knudsen number is defined by
where \(l_{mfp}\) is the local mean free path. The reference Knudsen number \({Kn}_{ref}\) is included as a critical value to evaluate the equilibrium and nonequilibrium regimes and thus directly control the waveparticle decomposition in each control volume. Then, the analytical free transport flux from unsampled particles is amended as
where
In the AUGKWP method, a hyperbolic tangent function shown in Fig.Â 3 is used in the determination of the waveparticle decomposition due to its boundness, smoothness, and convexity. Its boundness ensures a natural transition without further restriction, and the smoothness avoids the oscillation in the transition regime. Its convexity satisfies the expectation for sampling particles with aÂ slow rate of change in the continuum regime and a large rate otherwise. Moreover, the reference Knudsen number \({Kn}_{ref}\) can be straightforwardly added in this function as a value to distinguish the flow regime, and can be conveniently adjusted according to the flow condition.
FigureÂ 4 illustrates the waveparticle decomposition in the AUGKWP method in the limit of free molecular and continuum flowÂ regimes. Different from domain decomposition methods, the AUGKWP method employs an adaptive waveparticle formulation in each control volume with a unified treatment when the reference Knudsen number is fixed. In the AUGKWP method, the adjustment of theÂ reference Knudsen number influences the weights of wave and particles in each cell, instead of identifying different flow regimes according to this parameter in the conventional domaindecomposition approaches. In other words, the AUGKWP method has no buffer zones to distinguish and connect fluid and kinetic solvers. Additionally, since the gradientlength local Knudsen number is not influenced by the numerical resolution in a computational domain, it only identifies the local nonequilibrium state. The analytical wave will be mainly used in the equilibrium and nearequilibrium regimes whatever the cell resolution \((\Delta x, \Delta t)\) is adapted in AUGKWP, and the computational efficiency will be improved significantly.
5 Numerical validation
In this section, the AUGKWP method is tested in many cases. Since most of the cases are external flows, the determination of the initial condition of theÂ free stream at different Knudsen numbers will be provided here first. For a specific gas, the density in the free stream corresponding to a given Knudsen number is
where m is the molecular mass and \(L_{ref}\) is the reference length to define the Knudsen number. The dynamic viscosity is calculated from the translational temperature by the power law,
where \(\mu _{ref}\) is the reference dynamic viscosity at the temperature \(T_{ref}\).
In the tests, aÂ diatomic gas of nitrogen gas is employed with molecular mass \(m=4.65\times 10^{26}\) kg, \(\alpha =1.0\), \(\omega =0.74\), and the reference dynamic viscosity \(\mu _{ref} = 1.65\times 10^{5}\) \(\mathrm {Nsm^{2}}\) at the temperature \(T_{ref} = 273\) K. In the computations, the freestream or upstream values are used to nondimensionalize the flow variables, i.e.,
5.1 Sod shock tube test
The Sod shock tube problem is computed at different Knudsen numbers to verify the acceleration effect and capability for simulating the continuum and rarefied flows by the AUGKWP method. The nondimensional initial condition is
The spatial discretization is carried out by a onedimensional structured mesh with 200 uniform cells. The inlet and outlet of the tube are treated as far fields. The Courantâ€“Friedrichsâ€“Lewy (CFL) number is taken as 0.5. The critical value for waveparticle decomposition is chosen as \({Kn}_{ref} = 0.01\). The output time of the simulation is \(t = 0.12\).
The density, velocity and temperature obtained by the original UGKWP method, the AUGKWP method, and UGKS at different Knudsen numbers from \(10^{5}\) to 10 are plotted in Figs.Â 5, 6, 7, 8, 9, 10 and 11. The preset reference number of particles in both theÂ original UGKWP and AUGKWP methods is 400 per cell. Here, the purpose of setting this number of particles in computation is to show thatÂ the noise introduced in the AUGKWP method is acceptable compared with the original UGKWP method for unsteady flows when particles are not sufficiently enough. The results show the AUGKWP method can maintain the same solutions as the original UGKWP method and match with the UGKS solutions in all Knudsen regimes.
In these tests, a small time step determined by high spatial resolution leads to a large cell Knudsen number \({Kn}_c = \tau / \Delta t\). For the original UGKWP method, particles are sampled according to \(e^{\Delta t/ \tau }\) for the free transport particle with \(t_f = \Delta t\) even in the uniform equilibrium regions. For the AUGKWP method, the analytical wave formulation will be used in the equilibrium region. The numerical particle mass fraction given by the original UGKWP method and the AUGKWP method, the exponential function of theÂ cell Knudsen number \(e^{\Delta t/\tau }\), and the gradientlength local Knudsen number \({Kn}_{Gll}\) for theÂ Sod tube at \({Kn}= 10^{4}\) are plotted in Fig.Â 12. It shows the particles appear in the original UGKWP method even though the flow regime is continuum, while for the AUGKWP method, with the consideration of theÂ local Knudsen number, particles are sampled in the nonequilibrium region only, such as the shock front region with \({Kn}_{Gll} > 0.01\). Therefore, high computational efficiency is achieved in AUGKWP. TableÂ 1 is the comparison of the simulation times between the original UGKWP and AUGKWP methods at different Knudsen numbers. It shows theÂ AUGKWP method fully recovers the hydrodynamical solver without using particles in the equilibrium flow regime regardless ofÂ the cell resolution used.
5.2 Shock structure
To validate the capability of theÂ AUGKWP method for describing aÂ strong nonequilibrium state, shock structure at upstream Mach number \({Ma} = 4\) and 10 is investigated. The computational domain \([25, 25]\) has a length of 50 times of the particle mean free path and is divided by 100 cells uniformly. The left and right boundaries are treated as far field conditions. The CFL number takes 0.5. The reference Knudsen number is \({Kn}_{ref} = 0.001\). In this study, the upstream temperature is \(T_1 = 50\) K. The rest of theÂ parameters could be obtained from the nondimensional initial condition.
In theÂ kinetic theory, the particle collision time depends on the particle velocity. In order to cope with this physical reality, the relaxation time of the highspeed particles is amended in the UGKWP and AUGKWP methods [53],
with two parameters \(a= 0.1\) and \(b=5\).
For the UGKWP and AUGKWP methods, 400 simulation particles are used in each cell. To reduce the statistical noise, the timeaveraging processÂ is taken from theÂ 2500th step over 12500 steps. The normalized density and temperature from the original UGKWP method, the AUGKWP method, and the UGKS are plotted in Figs.Â 13â€“14. The results from the current method have good agreement with that from the original UGKWPÂ method and the deterministic method.
5.3 Flow around a circular cylinder
Highspeed flow passing over a semicircular cylinder at a Mach number ofÂ 15 and \({Kn} = 0.001\) is simulated. The diameter of the cylinder is \(D = 0.08\) m. The Knudsen number is defined with respect to the diameter. The computational domain is discretized by \(280 \times 200 \times 1\) quadrilateral cells. The initial reference number of particles \(N_r\) is set as 2000. The initial temperature of theÂ free stream gives \(T_\infty = 217.5\) K, and the isothermal wall temperature is fixed at \(T_w = 1000\) K. The CFL number is 0.5. The reference Knudsen number is \({Kn}_{ref} = 0.01\). FigureÂ 15 plots the contours of theÂ flow field computed by the AUGKWP method, where an initial flow field provided by 10000 steps of theÂ GKS calculation [51] is adopted, and 60000 steps of the AUGKWP and UGKWP methods have been carried out to achieve a steady state. FigureÂ 16 shows the comparison between the AUGKWP methodÂ and theÂ original UGKWP method for the density, velocity in theÂ x direction, and temperature extracted along the \(45^{\circ }\) line in the upstream. The results from theÂ AUGKWP method are reasonable. The numerical particle mass fraction in Fig.Â 17 implies the efficiency of the new waveparticle decomposition method. The simulation times for the AUGKWP and UGKWP methods on Tianhe2 with 3 nodes (72 cores, Intel Xeon E52692 v2, 2.2 GHz) are 15623 s and 30487 s, respectively.
5.4 Flow around an Apollo reentryÂ capsule
Hypersonic flow at \({Ma} = 5\) and \({Kn} = 10^{3}\) passing over an Apollo reentryÂ capsule in the transition flow regime is simulated for nitrogen gas. This case shows the efficiency and capability of the AUGKWP method for simulating the largescale threedimensional hypersonic flow. The reference length for the definition of the Knudsen number is \(L_{ref} = 3.912\) m. As shown in Fig.Â 18, the mesh consists of 372500 cells. The reference Knudsen number is set as \({Kn}_{ref} = 0.01\) and the initial reference number of particles \(N_r\) is 400 per cell. The flow at free stream has an initial temperature \(T_\infty = 142.2\) K, and the reentry surface is treated as anÂ isothermal wall with a constant temperature \(T_w = 500\) K. The angle of attack is \(30^{\circ }\). The initial flow field is prepared by theÂ GKS calculation with 10000 steps. Then, 25000 steps have been carried out to achieve a steady state solution.
FigureÂ 19 shows the distributions of temperature, pressure, local Knudsen number and Mach number computed by theÂ AUGKWP method. The quantitative comparison between the AUGKWP methodÂ and theÂ original UGKWP method for theÂ density, velocity in theÂ x direction, and temperature extracted from theÂ central axis along the windward is plotted in Fig.Â 20. Good agreement has been obtained. The efficiency of theÂ AUGKWPÂ method has been much improved due to its well controlled mass fraction of particles, as shown in Fig.Â 21. Particles in the nonequilibrium flowÂ are presented. The computational times for the AUGKWP and UGKWP methods are 24545 s and 42154 s on Tianhe2 with 5 nodes (120 cores, Intel Xeon E52692 v2, 2.2 GHz). The AUGKWPÂ method becomes an indispensable tool for largescale threedimensional hypersonic rarefied flow simulations.
5.5 Nozzle plume flow into a background vacuum
In this case, the AUGKWP method is applied to the \(\textrm{CO}_2\) expansions into a background vacuum. The unsteady and multiscale process of this plume flow is hard to compute with acceptive accuracy by conventional DSMCCFD hybrid methods with timedependent buffer zones, especially in the initial flow expansion stage. The UGKS, as a multiscale method, is hard to simulate such a flow with the requirement of covering a wide range ofÂ discretized particle velocity space to capture the high Mach number jet in theÂ flow acceleration process through the nozzle. The AUGKWP method treats the multiscale and multispeed expansion flow systematically through the dynamically adaptive waveparticle decomposition in each numerical cell, and makes the simulation acceptable in the memory requirement.
The geometry of the nozzle and the twodimensional mesh with 47186 cells are used in the simulation, as shown in Fig.Â 22. The inlet boundary condition is set with a temperature 710 K and pressure 36.5 torr. The gas \(\textrm{CO}_2\) is employed with the molecular mass \(m=7.31 \times 10^{26}\) kg, the ratio of specific heats \(\gamma = 1.4\), and \(\omega = 0.67\). The nozzle wall is treated as anÂ isothermal one with \(T_w = 300\) K. The background environment is set with a temperature \(T_B = 300\) KÂ and low pressure \(P_B = 0.01\) Pa. The CFL number is 0.5. The reference number of particles is \(N_r=400\). For the AUGKWP method, the reference Knudsen number is \({Kn}_{ref} = 0.01\).
The simulation covers the whole gas expansion process into a background vacuum through three stages, such as the initial, developing, and steady stages. FiguresÂ 23, 24 and 25 show the distributions of temperature, gradientlength dependent local Knudsen number, and Mach number along the streamline, and numerical particle mass fraction of the AUGKWP and UGKWP methods in each stage.
The flow field in the initial stage is shown in Fig.Â 23. Particles with aÂ large mean free path transport first to the background vacuum. The expansion gas forms a nonequilibrium central region. The AUGKWP method employs particles in this highly expanded region only, while the analytical wave is used in other regions. However, the UGKWP methodÂ adapts particles everywhere, even in the uniformly undisturbed background equilibrium region.
In the developing stage (see Fig.Â 24), a continuum flow regime appears near the nozzle exit, a transition regime forms around the high temperature expansion region, and a free molecular flow remains in the front of the plume. The gradientlength local Knudsen number shows a variation with 12 orders of magnitude in the whole computational domain. The simulation of this unsteady multiscale transport requires a method with the capability of capturing continuum and rarefied flows simultaneously at any moment by following the plume flow. The unified treatment in theÂ AUGKWPÂ method with a waveparticle decomposition in each control volume allows an instant and adaptive description in each cell in this expansion process.
FigureÂ 25 presents the plume flow approaching a steady state, where the gas is fully expanded with a steady flow pattern. The AUGKWP method provides a clear separation of different flow regimes, i.e., a continuum flow in the expansion region, a free molecular flow in the background flow, and a transition flow between them (see Fig.Â 25(c)). In the original UGKWPÂ method, the numerical time step determined by the smallest cell size leads the particle representation in almost theÂ whole computational domain. The quantitative comparison of theÂ density, velocity inÂ the x direction, and temperature along the centerline at the steady stage is plotted in Fig.Â 26. It shows agreement between the results given by theÂ AUGKWP and UGKWP methods.
Overall, the AUGKWP method gives a nonequilibrium state guided waveparticle decomposition. The computational times in the current studies on Tianhe2 with 20 nodes (480 cores) are 12.6 hours and 36.1 hours for theÂ AUGKWP and UGKWP methods, respectively.
6 Conclusion
The UGKWPÂ method is a multiscale method for flow simulations in all regimes. The UGKWP method adopts a waveparticle decomposition to recover the multiscale transport uniformly in each control volume. In this paper, an adaptive unified gaskinetic waveparticle (AUGKWP) method is developed to further optimize the waveparticle decomposition in the original UGKWP. In order to concentrate particles in the nonequilibrium region only, instead of using the cell Knudsen number \({Kn}_c = \tau / \Delta t\) only in the original UGKWP method, the AUGKWP method introduces a flow gradientrelated local Knudsen number as well for the decomposition of wave and particle. As a result, the AUGKWPÂ method avoids using particles in the highly dilute background equilibrium region and in the continuum flow simulation with the use of anÂ extremely small numerical time step. The AUGKWP method provides a physically reliable waveparticle decomposition and guarantees the appearance of particles in the nonequilibrium region only, regardless of the mesh resolution. Many test cases are used to validate the efficiency and accuracy of the AUGKWP method. In comparison with the original UGKWP method, due to theÂ significant reduction of particles in the AUGKWPÂ method, the scheme can speed up theÂ computation, reduce theÂ memory requirement, and maintain the same solution accuracy as the original multiscale waveparticle method. The AUGKWP methodÂ will become a useful and indispensable tool in the simulation of highspeed rarefied and continuum flows in aerodynamic applications.
Availability of data and materials
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Bird GA (1994) Molecular gas dynamics and the direct simulation of gas flows, 2nd edn. Clarendon Press, Oxford
Xu K (2021) A unified computational fluid dynamics framework from rarefied to continuum regimes. Cambridge University Press, Oxford
Fan J, Shen C (2001) Statistical simulation of lowspeed rarefied gas flows. J Comput Phys 167(2):393â€“412
Shen C (2005) Rarefied gas dynamics: fundamentals, simulations and micro flows. Springer Berlin, Heidelberg
Sun Q, Boyd ID (2002) A direct simulation method for subsonic, microscale gas flows. J Comput Phys 179(2):400â€“425
Baker LL, Hadjiconstantinou NG (2005) Variance reduction for Monte Carlo solutions of the Boltzmann equation. Phys Fluids 17(5):051703
Homolle TM, Hadjiconstantinou NG (2007) A lowvariance deviational simulation Monte Carlo for the Boltzmann equation. J Comput Phys 226(2):2341â€“2358
Degond P, Dimarco G, Pareschi L (2011) The momentguided Monte Carlo method. Int J Numer Methods Fluids 67(2):189â€“213
Pareschi L, Russo G (2000) Asymptotic preserving Monte Carlo methods for the Boltzmann equation. Transp Theory Stat Phys 29(3â€“5):415â€“430
Ren W, Liu H, Jin S (2014) An asymptoticpreserving Monte Carlo method for the Boltzmann equation. J Comput Phys 276:380â€“404
Dimarco G, Pareschi L (2011) Exponential RungeKutta methods for stiff kinetic equations. SIAM J Numer Anal 49(5):2057â€“2077
Chu CK (1965) Kinetictheoretic description of the formation of a shock wave. Phys Fluids 8(1):12â€“22
Yang JY, Huang JC (1995) Rarefied flow computations using nonlinear model Boltzmann equations. J Comput Phys 120(2):323â€“339
Mieussens L (2000) Discretevelocity models and numerical schemes for the BoltzmannBGK equation in plane and axisymmetric geometries. J Comput Phys 162(2):429â€“466
Tcheremissine F (2005) Direct numerical solution of the Boltzmann equation. AIP Conf Proc 762(1):677â€“685
Kolobov VI, Arslanbekov RR, Aristov VV et al (2007) Unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement. J Comput Phys 223(2):589â€“608
Li Z, Zhang H (2009) Gaskinetic numerical studies of threedimensional complex flows on spacecraft reentry. J Comput Phys 228(4):1116â€“1138
Xu K, Huang J (2010) A unified gaskinetic scheme for continuum and rarefied flows. J Comput Phys 229(20):7747â€“7764
Wu L, Zhang J, Reese JM et al (2015) A fast spectral method for the Boltzmann equation for monatomic gas mixtures. J Comput Phys 298:602â€“621
Aristov VV (2001) Direct methods for solving the Boltzmann equation and study of nonequilibrium flows. Springer, Dordrecht
Li Z, Zhang H (2004) Study on gas kinetic unified algorithm for flows from rarefied transition to continuum. J Comput Phys 193(2):708â€“738
Li Z, Peng A, Ma Q et al (2019) Gaskinetic unified algorithm for computable modeling of Boltzmann equation and application to aerothermodynamics for falling disintegration of uncontrolled TiangongNo. 1 spacecraft. Adv Aerodyn 1(1):4
Guo Z, Xu K, Wang R (2013) Discrete unified gas kinetic scheme for all Knudsen number flows: Lowspeed isothermal case. Phys Rev E 88(3):033305
Chen S, Zhang C, Zhu L et al (2017) A unified implicit scheme for kinetic model equations. Part I. Memory reduction technique. Sci Bull 62(2):119â€“129
Chen S, Xu K (2015) A comparative study of an asymptotic preserving scheme and unified gaskinetic scheme in continuum flow limit. J Comput Phys 288:52â€“65
Zhu Y, Zhong C, Xu K (2016) Implicit unified gaskinetic scheme for steady state solutions in all flow regimes. J Comput Phys 315:16â€“38
Zhu Y, Zhong C, Xu K (2017) Unified gaskinetic scheme with multigrid convergence for rarefied flow study. Phys Fluids 29(9):096102
Zhu Y, Zhong C, Xu K (2019) An implicit unified gaskinetic scheme for unsteady flow in all Knudsen regimes. J Comput Phys 386:190â€“217
Jiang D, Mao M, Li J et al (2019) An implicit parallel UGKS solver for flows covering various regimes. Adv Aerodyn 1(1):8
Chen S, Xu K, Lee C et al (2012) A unified gas kinetic scheme with moving mesh and velocity space adaptation. J Comput Phys 231(20):6643â€“6664
Mouhot C, Pareschi L (2006) Fast algorithms for computing the Boltzmann collision operator. Math Comput 75(256):1833â€“1852
Wu L, White C, Scanlon TJ et al (2013) Deterministic numerical solutions of the Boltzmann equation using the fast spectral method. J Comput Phys 250:27â€“52
Filbet F, Jin S (2010) A class of asymptoticpreserving schemes for kinetic equations and related problems with stiff sources. J Comput Phys 229(20):7625â€“7648
Dimarco G, Pareschi L (2013) Asymptotic preserving implicitexplicit RungeKutta methods for nonlinear kinetic equations. SIAM J Numer Anal 51(2):1064â€“1087
Guo Z, Li J, Xu K (2023) Unified preserving properties of kinetic schemes. Phys Rev E 107(2):025301
Li Z, Fang M, Jiang X et al (2013) Convergence proof of the DSMC method and the gaskinetic unified algorithm for the Boltzmann equation. Sci China Phys Mech Astron 56:404â€“417
Guo Z, Xu K (2021) Progress of discrete unified gaskinetic scheme for multiscale flows. Adv Aerodyn 3(1):6
Fei F, Zhang J, Li J et al (2020) A unified stochastic particle BhatnagarGrossKrook method for multiscale gas flows. J Comput Phys 400:108972
Fei F, Ma Y, Wu J et al (2021) An efficient algorithm of the unified stochastic particle BhatnagarGrossKrook method for the simulation of multiscale gas flows. Adv Aerodyn 3(1):18
Liu C, Zhu Y, Xu K (2020) Unified gaskinetic waveparticle methods I: Continuum and rarefied gas flow. J Comput Phys 401:108977
Zhu Y, Liu C, Zhong C et al (2019) Unified gaskinetic waveparticle methods. II. Multiscale simulation on unstructured mesh. Phys Fluids 31(6):067105
Xu K (2015) Direct modeling for computational fluid dynamics: Construction and application of unified gaskinetic scheme. World Scientific, Hong Kong
Liu S, Zhong C, Fang M (2020) Simplified unified waveparticle method with quantified modelcompetition mechanism for numerical calculation of multiscale flows. Phys Rev E 102(1):013304
Yang LM, Li ZH, Shu C et al (2022) Discrete unified gaskinetic waveparticle method for flows in all flow regimes. Phys Rev E 108(1):015302
Xu X, Chen Y, Liu C et al (2021) Unified gaskinetic waveparticle methods V: Diatomic molecular flow. J Comput Phys 442:110496
Wei Y, Zhu Y, Xu K (2022) Unified gaskinetic waveparticle methods VII: Diatomic gas with rotational and vibrational nonequilibrium. arXiv preprint arXiv:2211.12922
Li W, Liu C, Zhu Y et al (2020) Unified gaskinetic waveparticle methods III: Multiscale photon transport. J Comput Phys 408:109280
Liu C, Xu K (2021) Unified gaskinetic waveparticle methods IV: Multispecies gas mixture and plasma transport. Adv Aerodyn 3(1):9
Yang X, Wei Y, Shyy W et al (2022) Unified gaskinetic waveparticle method for threedimensional simulation of gasparticle fluidized bed. Chem Eng J 453:139541
Bhatnagar PL, Gross EP, Krook M (1954) A model for collision processes in gases. I. Small amplitude processes in charged and neutral onecomponent systems. Phys Rev 94(3):511â€“525
Xu K (2001) A gaskinetic BGK scheme for the NavierStokes equations and its connection with artificial dissipation and Godunov method. J Comput Phys 171(1):289â€“335
Venkatakrishnan V (1995) Convergence to steady state solutions of the Euler equations on unstructured grids with limiters. J Comput Phys 118(1):120â€“130
Xu X, Chen Y, Xu K (2021) Modeling and computation for nonequilibrium gas dynamics: Beyond single relaxation time kinetic models. Phys Fluids 33(1):011703
Acknowledgements
The CFD group members at HKUST are deserving of the authorsâ€™ gratitude for their weekly discussions and collaborative efforts.
Funding
This work was supported by theÂ National Key R&D Program of China (Grant No. 2022YFA1004500), National Natural Science Foundation of China (Grant No.Â 12172316), and theÂ Hong Kong Research Grants Council (Grant Nos.Â 16208021,Â 16301222).
Author information
Authors and Affiliations
Contributions
All authors contributed equally to this work.
Corresponding author
Ethics declarations
Competing interests
Not applicable.
Additional information
Publisherâ€™s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Wei, Y., Cao, J., Ji, X. et al. Adaptive waveparticle decomposition in UGKWP method for highspeed flow simulations. Adv. Aerodyn. 5, 25 (2023). https://doi.org/10.1186/s4277402300156y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s4277402300156y