US2019354650A1PendingUtilityA1
Analytical Tools and Method for Modeling Transport Processes in Fluids
Est. expiryJan 19, 2038(~11.5 yrs left)· nominal 20-yr term from priority
Inventors:Nikolai Kislov
G06F 30/23G06F 2111/10G06F 17/13G06F 17/18G06F 2217/16G06F 17/5018
52
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Claims
Abstract
An improved computer implemented method for modeling transport processes in fluids is disclosed. The method is adapted to model gas flow including dilute gas flow at high Knudsen numbers. Instead of modeling based on using an infinitesimal fluid element of a continuous medium, the method approximates fluid flow as a model gas flow. The method is based on assuming that each of a plurality of particles, which compose the model gas, travels with a probability between any of two points in space occupied by the model gas by following a ballistic trajectory governed by a law of motion in free space and treating each of the plurality of ballistic particles as a property carrier transporting one or more of mass, momentum, and energy between the points of consecutive collisions. The major steps of the method are indicated in the accompanying flow chart. The method also delivers a new basis for prediction of dynamic evolution of the model gas system by considering a pre-established or known dynamic history of the system during a pre-initial period.
Claims
exact text as granted — not AI-modified1 . A computer implemented method for modeling transport processes in fluids comprising:
approximating fluid flow in a fluid system as a flow of a model gas in a model gas system being identical to the fluid system; assuming that each of a plurality of particles, which compose the model gas, travels with a probability between any points in a space occupied by the model gas by following a ballistic trajectory governed by a law of motion in free space, the ballistic trajectory having a starting point in one of a plurality of points of original collisions and an ending point in one of a plurality of points of ending collisions; treating each of the plurality of points of original collisions as a point source; treating each of the plurality of points of ending collisions as a point sink; treating each of a plurality of ballistic particles moving from the point source to the point sink as a property carrier created in the point source at a time of an original collision by obtaining one or more of properties comprising one or more of mass, momentum, and energy of specific values being intrinsic to the point source, and ended in the point sink at the time of an ending collision by transferring one or more of properties of specific values in the point sink; forming an integral property balance equation for each of one or more properties being transported by the plurality of ballistic particles in each of a plurality of non-moving points at a given time, wherein the plurality of ballistic particles in each of the plurality of non-moving points at the given time comprises a plurality of converging ballistic particles and a plurality of diverging ballistic particles, and wherein each of the plurality of non-moving points is treated as the point sink for each of the plurality of converging ballistic particles and as the point source for each of the plurality of diverging ballistic particles; and
1 . A computer implemented method for modeling transport processes in fluids comprising:
using a fluid model, wherein the fluid model comprises treating a fluid flow including a flow of rarefied gases in a fluid system as a flow of a model gas in a model gas system being identical to the fluid system, wherein the model gas is composed of a plurality of particles including molecules, which move randomly and interact by collisions, wherein each of the plurality of particles is assigned to travel with a probability between any points in a space occupied by the model gas by following a ballistic trajectory governed by a law of motion, the ballistic trajectory having a starting point in a point of original collision and an ending point in a point of ending collision, wherein each of the plurality of points of original collision is treated as a point source, wherein each of the plurality of points of ending collision is treated as a point sink, and wherein each of a plurality of ballistic particles moving from the point source to the point sink is treated as a property carrier created in the point source at a time of an original collision by obtaining one or more of properties comprising one or more of mass, momentum, and energy of specific values being intrinsic to the point source, and ended in the point sink at a time of an ending collision by transferring one or more of properties of specific values in the point sink; and using a computer for simulating the fluid flow, wherein the computer for simulating the fluid flow comprises a computer readable medium for storing data associated with a computer program, and a processor being connected for accessing the computer program and operable to run the computer, wherein the computer program is configured to use the fluid model and a model of property balance, wherein the model of property balance in a combination the fluid model transforms parameters characterizing motion of a plurality of ballistic particles into parameters characterizing the fluid flow, wherein the model of property balance comprises establishing an integral property balance equation for a property being transported by a plurality of converging ballistic particles and a plurality of diverging ballistic particles in each of a plurality of non-moving points in the space occupied by the model gas at a given time, and wherein the computer program is operable to run on the computer for simulating the fluid flow to compute the flow of the model gas in each of the plurality of non-moving points at the given time and to display results of computing, thereby reliably predicting the fluid flow including the flow of rarefied gases.
2 . The method of claim 1 , wherein a value of property delivered in the point sink by each of the plurality of converging ballistic particles is evaluated regarding whether the value of property is conserved, whether the value of property is changed because of aging, and whether the value of property is modified because of interaction with an external field during a ballistic traveling time.
3 . The method of claim 1 , wherein
each of a plurality of collisions on a gas-solid interface of the model gas system, which has resulted in diffuse particle scattering from the gas-solid interface, is treated as an act of interaction involving a property transfer from the gas-solid interface to a scattered particle, wherein each of a plurality of points of diffuse particle scattering on the gas-solid interface is treated as a heterogeneous point source for each of a plurality of scattered particles, wherein velocity of each of a plurality of heterogeneous point sources on the gas-solid interface is assigned to be equal to the velocity of the gas-solid interface in each of a plurality of corresponding points of diffuse particle scattering at time of diffuse particle scattering, and wherein point source strength of each of the plurality of heterogeneous point sources on the gas-solid interface is assigned to be directly proportional to a property accommodation coefficient in each of the plurality of corresponding points of diffuse particle scattering at the time of diffuse particle scattering, the property accommodation coefficient which is the probability, for an incident particle, to accommodate one or more of properties intrinsic to the gas-solid interface and to scatter back in the model gas, the property accommodation coefficient being in a range from zero to one.
4 . The method of claim 1 , further comprising steps of:
establishing an initial time, wherein the initial time is treated as the time of a specific modification of the model gas system in a specific location comprising specific modification of one or more of temperature, velocity of a gas-solid interface, and application of an external field that may modify a property value; specifying the given time, wherein the given time is greater than or equal to the initial time; specifying the model gas system, wherein the model gas system is specified by defining model gas properties comprising a gas material, pressure, temperature and by establishing a geometry model, wherein establishing the geometry model comprises setting geometry and boundary conditions during a period from a pre-initial time until the given time; establishing discretization parameters comprising a set of discretization points in the space occupied by the model gas, wherein each of the set of discretization points is assigned to a corresponding point of the plurality of non-moving points; calculating, for each of the set of discretization points in the space occupied by the model gas, a local initial time, wherein the local initial time is greater than or equal to the initial time; obtaining, for each of the set of discretization points in the space occupied by the model gas at a given advanced time, which is ahead of the local initial time, a past property value; formulating, for each of the set of discretization points at the given time, an approximated integral form of property balance, wherein the approximated integral form of property balance is established in a way that, in a selected point of the set of discretization points at the given time, a net rate of property influx per unit volume from the model gas system, which is formed by a set of converging particles and calculated by summing a rate of property influx from the set of discretization points surrounding the selected point is equated to a net rate of property efflux per unit volume from the selected point of the set of discretization points, separating the plurality of converging ballistic particles on a first plurality of converging ballistic particles and on a second plurality of converging ballistic particles, where each of the first plurality of converging ballistic particles delivers one or more of past property values from the model gas system, where each of the second plurality of converging ballistic particles delivers one or more of present property values from the model gas system, where each of past property values is obtained from one of the plurality of points of original collisions at the time ahead of the local initial time, where each of past property values is well defined in each of the plurality of points of original collision at time of original collision, where each of present property values is obtained from one of the plurality of points of original collisions at the time which follows the local initial time, and where a present property value in each of the plurality of points of the original collision is treated as an unknown property value; and computing the flow of the model gas by considering an impact of a plurality of initial converging ballistic particles and taking into account an effect of mutual dependence of property exchange between particles interacting by collisions, which comprises a sequential approximation method.
5 . The method of claim 1 , wherein the integral property balance equation is established in a way that, in a general point of the plurality of non-moving points in the space occupied by the model gas at the given time, a net rate of property influx per unit volume, which is formed by the plurality of converging ballistic particles, each of the plurality of converging ballistic particles is selected from the plurality of ballistic particles by the ballistic trajectory having the ending point in the general point at the given time, is equated to sum a temporal rate of a property change per unit volume and a net rate of property efflux per unit volume, which is formed by the plurality of diverging ballistic particles, each of the plurality of diverging ballistic particles is selected from the plurality of ballistic particles by the ballistic trajectory having the starting point in the general point at the given time.
6 . A computer implemented method for modeling transport processes in fluids comprising:
using a fluid model, wherein using the fluid model comprises treating a fluid flow including a flow of rarefied gases in a fluid system as a flow of a model gas in a model gas system being identical to the fluid system, wherein the model gas is composed of a plurality of particles, including molecules, which move randomly and interact by collisions, wherein each of the plurality of particles is assigned to travel with a probability between any of two points in a space occupied by the model gas by following a ballistic trajectory governed by a law of motion, the ballistic trajectory having a starting point in a point of original collision and an ending point in a point of ending collision, and wherein each of a plurality of ballistic particles transports a combination of one or more of properties comprising one or more of mass, momentum, and energy from the starting point to the ending point; using a model of property balance, wherein the model of property balance in a combination the fluid model transforms parameters characterizing motion of a plurality of ballistic particles into parameters characterizing the fluid flow, wherein the model of property balance comprises: specifying a geometry model, defining a net rate of property influx per unit volume in a general non-moving point in the space occupied by the model gas at a given time, the net rate of property influx per unit volume, which is formed by a plurality of converging ballistic particles, each of the plurality of converging ballistic particles is selected from the plurality of ballistic particles by the ballistic trajectory having the ending point in the general non-moving point at the given time, defining a net rate of property efflux per unit volume from the general non-moving point at the given time, the net rate of property efflux per unit volume, which is formed by a plurality of diverging ballistic particles, each of the plurality of diverging ballistic particles is selected from the plurality of ballistic particles by the ballistic trajectory having the starting point in the general non-moving point at the given time, and establishing an integral property balance equation for each of one or more properties being transported by the plurality of ballistic particles in the general non-moving point; and using a computer for simulating the fluid flow, wherein the computer for simulating the fluid flow comprises a computer readable medium for storing data associated with a computer program, and a processor for executing a sequence of instructions from the computer program, wherein the computer program is configured to use the fluid model and the model of property balance, and wherein the computer program is operable to run on the computer for simulating the fluid flow to compute the flow of the model gas in every general point of a plurality of non-moving points in the space occupied by the model gas at the given time and to display results of computing, thereby reliably predicting the fluid flow including the flow of rarefied gases.
7 . The method of claim 6 , wherein the step of specifying the geometry model further comprises:
establishing geometry and boundary conditions, wherein geometry and boundary conditions are established during a period from a pre-initial time until the given time; and establishing a dynamic history of the model gas, wherein the dynamic history of the model gas, which is in form of one or more of instant distribution patterns of model gas properties within the model gas system at a set of different sequential moments in time, is established during a pre-initial period, and wherein the pre-initial period is set to be not less than a reliable period before an initial time, wherein the initial time is treated as a time before which the fluid system is described,
8 . The method of claim 7 , wherein the reliable period before the initial time is estimated as
ϕ
i
0
rel
≅
-
ln
(
MAV
)
[
P
c
v
rel
]
m
,
where φ i0 rel is the reliable period before the initial time, MAV is a minimum absolute value corresponding to a precision format supported by a computer hardware, P c is a number of particles within a collision tube of a unit length placed in the model gas, v rel is an average magnitude of velocity of a traveling particle with respect to a nearby particle, and [P c v rel ] max is a highest value of product of P c v rel in the model gas system.
9 . The method of claim 6 , wherein the plurality of converging ballistic particles comprises a first plurality of converging ballistic particles,
where the ballistic trajectory of one of the first plurality of converging ballistic particles has the starting point in the space occupied by the model gas, wherein the step of defining the net rate of property influx per unit volume from the model gas system further comprises a step of defining a net rate of total property influx per unit volume from a surrounding model gas, wherein the net rate of total property influx per unit volume is formed by a first plurality of converging ballistic particles, each of the first plurality of converging ballistic particles is selected from the plurality of converging ballistic particles by the ballistic trajectory having the starting point in one of a plurality of points of original collisions within the space occupied by the model gas, and wherein each of the plurality of points of original collisions is treated as a point source.
10 . The method of claim 9 , wherein the step of defining the net rate of total property influx per unit volume from the surrounding model gas comprises:
identifying, for each of the first plurality of converging ballistic particles, the ballistic trajectory; defining the probability of traveling along the ballistic trajectory; defining a net rate of particle efflux per unit volume from one of the plurality of points of an original collision at time of the original collision, wherein the time of the original collision for each of the plurality of converging ballistic particles is identified by the ballistic trajectory, and wherein velocity of the point source for each of the first plurality of converging ballistic particles is assigned to be equal to mass flow velocity of the model gas in a corresponding point of the original collision at the time of the original collision; defining a net rate of property flux per unit area in the general point at the given time from one of a plurality of point sources; defining a net rate of total property flux per unit area in the general point at the given time from the plurality of point sources; and defining the net rate of total property influx per unit volume in the general point at the given time by applying divergence operator to the net rate of total property flux.
11 . The method of claim 10 , wherein the step of the identifying the ballistic trajectory comprises:
determining one or more of characteristics of the ballistic trajectory comprising the time of the original collision, an appropriate instant unit vector defining the ballistic trajectory in the starting point, and a velocity vector in the ending point of the ballistic trajectory at the given time; and defining a size of an expansion zone.
12 . The method of claim 10 , wherein the step of defining the net rate of total property influx per unit volume in a one-dimensional configuration further comprises a step of formulating the net rate of total property influx as
B
in
Ψ
λ
_
F
(
y
,
t
)
=
-
1
2
∂
∂
y
∫
yb
2
yb
1
Q
i
(
t
,
t
i
′
)
Z
V
(
t
i
′
,
y
′
)
v
(
t
i
′
,
y
′
,
t
,
y
)
v
T
(
t
i
′
,
y
′
)
Ψ
in
(
t
i
′
,
y
′
,
t
,
y
)
dy
′
,
wherein
∂
∂
y
is the divergence operator in one-dimensional configuration,
B in Ψλ_F is the net rate of property influx per unit volume,
t is the given time,
t i ′ is the time of the original collision,
y is position of the ending point in one of the plurality of points of ending collisions,
y′ is position of the starting point in one of the plurality of points of original collisions,
v T (t i ′,y′) is average magnitude of thermal velocity in the starting point at time of original collision,
Z V (t i ′,y′) is rate of collisions per unit volume in the starting point at the time of original collision,
v(t i ′,y′,t,y) is a velocity vector in the ending point at the given time,
Q i (t,t i ′) is the probability of traveling along the ballistic trajectory,
Ψ in (t i ′,y′,t,y) is a property value delivered in the ending point at the given time by one of the first plurality of converging ballistic particles,
yb1 is a lower limit of integration over volume of space occupied by the model gas,
yb2 is an upper limit of integration over volume of space occupied by the model gas, and
Z V (t i ,y′) is calculated as
Z v ( t i ′,y ′)=½ P c ( t i ′,y ′) v rel ( t i ′,y ′) n ( t i ′,y ′),
where n(t i ′,y′) is particle density, P c (t i ′,y′) is an average number of collisions per unit length, and v rel (t i ′,y′) is an average magnitude of relative velocity.
13 . The method of claim 12 , wherein the probability of traveling along the ballistic trajectory in one-dimensional configuration is calculated as
Q i ( t,t i ′)=exp(−∫ t i ′ t P c ( {tilde over (y)} ( {tilde over (t)} )) v rel ( {tilde over (y)} ( {tilde over (t)} )) d{tilde over (t)} ),
where {tilde over (t)} is a parametric time in a range of t i ′<{tilde over (t)}≤t, {tilde over (y)}({tilde over (t)}) is a trajectory point of the ballistic trajectory at parametric time {tilde over (t)}, v rel ({tilde over (y)}({tilde over (t)})) is the average magnitude of relative velocity in the trajectory point, and P c ({tilde over (y)}({tilde over (t)})) is average number of collisions per unit length in the trajectory point.
14 . The method of claim 13 , wherein the average magnitude of relative velocity in the trajectory point is calculated by steps of:
defining an instant magnitude of velocity of one of the plurality of converging ballistic particles in the trajectory point with respect to a nearby particle in the trajectory point; and averaging the instant magnitude of velocity over a plurality of directions of a thermal velocity component of a nearby particle.
15 . The method of claim 6 , wherein a property value delivered in the ending point by each of the plurality of converging ballistic particles is calculated as
Ψ in =Ψ λ +Ψ g ,
where Ψ in is the property value delivered in the ending point, Ψ g is a field function characterizing property value of known measure, which is changed during traveling time, φ, because of interaction with external field comprising modification of momentum in gravitation field, Ψ λ is an age function characterizing the property value in the ending point of the ballistic trajectory, which is calculated as
Ψ λ =Ψ 0 e −λφ ,
where Δ is an aging coefficient, which is in range from negative to positive values including zero value, and where Ψ 0 is a starting function characterizing the property value carried by each of the plurality of ballistic particles in the starting point of the ballistic trajectory at a starting time, t i , which is calculated as
Ψ 0 ( t i ′,y ′)=Ψ 0− [ t i <t i0 ]+Ψ[ t i ≥t i0 ],
where t i0 is a local initial time, each of expressions with square brackets is Iverson bracket, which converts a statement in brackets into a number, the number is one if the statement is satisfied, and the number is zero otherwise, Ψ 0− is a pre-established property value in each of the plurality of points in space of the model gas system at the starting time ahead of the local initial time, and Ψ is a present property value in each of the plurality of points in space of the model gas system at the starting time greater than or equal to the local initial time.
16 . (canceled)
17 . The method of claim 6 , wherein the plurality of converging ballistic particles comprises a second plurality of converging ballistic particles,
where the ballistic trajectory of one of the second plurality of converging ballistic particles has the starting point on a gas-solid interface, wherein the step of defining the net rate of property influx per unit volume from the model gas system further comprises a step of defining a net rate of total property influx per unit volume from a gas-solid interface exhibiting mixed diffuse and specular particle scattering, wherein the net rate of total property influx per unit volume from the gas-solid interface is formed by way of a second plurality of converging ballistic particles, each of the second plurality of converging ballistic particles is selected from the plurality of converging ballistic particles by the ballistic trajectory having the starting point in one of a plurality of points of original collisions on the gas-solid interface, and wherein each of the plurality of points of original collisions resulted in diffuse particle scattering from the gas-solid interface is treated as a heterogeneous point source.
18 . The method of claim 17 , wherein the step of defining the net rate of total property influx per unit volume from the gas-solid interface comprises:
identifying the ballistic trajectory, for each of the second plurality of converging ballistic particles, wherein velocity of the heterogeneous point source for each of the second plurality of converging ballistic particles is assigned to be equal to the velocity of the gas-solid interface in a corresponding point of original collisions on the gas-solid interface at a time of diffuse particle scattering; defining the probability of traveling along the ballistic trajectory; defining a rate of collisions per unit area in one of a plurality of heterogeneous point sources on the gas-solid interface at the time of diffuse particle scattering, wherein the time of diffuse particle scattering for each of the second plurality of converging ballistic particles is identified by the ballistic trajectory; defining a net rate of property flux per unit area in the general point at the given time from one of the plurality of heterogeneous point sources; defining a net rate of total property flux in the general point at the given time from the plurality of heterogeneous point sources on one or more of gas-solid interfaces; and defining the net rate of total property influx per unit volume in the general point at the given time by applying divergence operator to the net rate of total property flux.
19 . The method of claim 18 , wherein the step of defining the rate of collisions comprises:
defining a bulk component of an impingement rate, which is formed by particles chosen from the plurality of converging ballistic particles originated from the original collisions in the space occupied by the model gas; defining a bulk-specular component of the impingement rate, which is formed by particles chosen from the plurality of converging ballistic particles originated from the original collisions in the space occupied by the model gas and having at least the last preceding specular scattering; defining a diffuse component of the impingement rate, which is formed by particles chosen from the plurality of converging ballistic particles originated from one or more of heterogeneous point sources; defining a diffuse-specular component of the impingement rate, which is formed by particles chosen from the plurality of converging ballistic particles originated from original diffuse scatterings from one or more of heterogeneous point sources and having at least the last preceding specular scattering; and summing the bulk component, the bulk-specular component, the diffuse component, and the diffuse-specular component of the impingement rate, wherein each of a plurality of impinging particles is selected by an impinging trajectory allowing to target the gas-solid interface in a specific location of the gas-solid interface at a specific time.
20 . The method of claim 18 , wherein the step of defining the net rate of total property influx per unit volume further comprises a step of formulating the net rate of total property influx as
B
in
Ψ
λ
_
b
=
-
∂
∂
y
[
Q
ib
(
t
,
t
ib
′
)
σ
Z
b
(
t
ib
′
,
y
b
)
v
b
(
t
ib
′
,
y
b
,
t
,
y
)
v
Tb
1
(
t
ib
′
,
y
b
)
Ψ
inb
(
t
ib
′
,
y
b
,
λ
,
ϕ
ib
1
)
]
,
wherein
B in Ψλ_b is the net rate of total property influx from the gas-solid interface in one-dimensional configuration,
t is the given time,
t ib ′ is a time of scattering from the gas-solid interface,
y is position of the ending point,
y b is position of the starting point in one of the plurality of points of original collisions on the gas-solid interface,
v Tb (t ib ′,y b ) is a magnitude of thermal velocity obtained in the starting point at the time of scattering from the gas-solid interface,
Z b (t ib ′,y b ) is the rate of collisions per unit area in the starting point at the time of scattering from the gas-solid interface,
v b (t ib ′,y b ,t,y) is a velocity vector in the ending point at the given time,
Q ib (t,t ib ) is the probability of traveling along the ballistic trajectory of the second plurality of converging ballistic trajectories,
Ψ inb (t ib ′,y′,λ,t,y) is a property value delivered in the ending point at the given time by one of the second plurality of converging ballistic particles,
λ is aging coefficient, and
σ is a property accommodation coefficient.
21 . The method of claim 6 , wherein the step of defining the net rate of property efflux per unit volume from the general point of the plurality of non-moving points at the given time in the model gas system comprises:
identifying, for each of the plurality of diverging ballistic particles, the ballistic trajectory starting from the general point of the plurality of non-moving points at the given time and ending in one of the plurality of points of ending collisions surrounding the general point; defining the probability of traveling along the ballistic trajectory; defining a vector field of a property flux per unit area in the ending point of the ballistic trajectory of a diverging ballistic particle; and defining the net rate of property efflux per unit volume by applying a divergence operator to the vector field of the property flux per unit area in the ending point; and shrinking volume of space surrounding the general point to the general point.
22 . The method of claim 21 , wherein the step of defining the net rate of property efflux per unit volume in one-dimensional configuration comprises a step of formulating the net rate of property efflux as
B
out
Ψ_
FS
(
t
,
y
)
=
1
2
{
∂
∂
y
[
n
(
t
,
y
)
Q
+
(
t
a
′
,
t
)
v
+
(
t
,
y
,
t
a
′
,
y
′
)
Ψ
λ
(
t
,
y
,
t
a
′
,
y
′
)
]
}
y
′
→
y
,
where
∂
∂
y
is the divergence operator in one-dimensional configuration,
B out Ψ_FS is the net rate of property efflux per unit volume,
t is the given time,
y is position of the starting point of a plurality of ballistic trajectories of diverging ballistic particles,
y′ is position of the ending point of one of the plurality of ballistic trajectories of diverging ballistic particles,
t a ′ is a time of positioning in the ending point,
v + (t,y,t a ′,y′) is a velocity vector of each of the plurality of diverging ballistic particle at the time of positioning the ending point,
Q + (t a ′,t) is the probability of traveling along the ballistic trajectory of one of the plurality of diverging ballistic particles,
Ψ λ (t,y,t a ′,y′) is property value carried by one of the plurality of diverging ballistic particles at the time of positioning the ending point, and
n(t,y) is particle density in the starting point at the given time.
23 . A computer implemented method for modeling transport processes in fluids being in a stable condition comprising:
using a fluid model, wherein using the fluid model comprises treating a fluid flow including a flow of rarefied gases in a fluid system as a flow of a model gas in a model gas system being identical to the fluid system, wherein the model gas is composed of a plurality of particles, including molecules, which move randomly and interact by collisions, wherein each of the plurality of particles is assigned to travel with a probability between any of two points in a space occupied by the model gas by following a ballistic trajectory governed by a law of motion, the ballistic trajectory having a starting point in a point of original collision and an ending point in a point of ending collision, and wherein each of a plurality of ballistic particles transports a combination of one or more of properties comprising one or more of mass, momentum, and energy from the starting point to the ending point; using a model of property balance, wherein the model of property balance in a combination the fluid model transforms parameters characterizing motion of a plurality of ballistic particles into parameters characterizing the fluid flow, wherein the model of property balance comprises: specifying geometry and boundary conditions, defining a net rate of property influx per unit volume in a general non-moving point in the space occupied by the model gas, the net rate of property influx per unit volume, which is formed by a plurality of converging ballistic particles, each of the plurality of converging ballistic particles is selected from the plurality of ballistic particles by the ballistic trajectory having the ending point in the general non-moving point, defining a net rate of property efflux per unit volume from the general non-moving point, the net rate of property efflux per unit volume, which is formed by a plurality of diverging ballistic particles, each of the plurality of diverging ballistic particles is selected from the plurality of ballistic particles by the ballistic trajectory having the starting point in the general non-moving point, and establishing an integral property balance equation for each of one or more properties being transported by the plurality of ballistic particles in the general non-moving point; and using a computer for simulating the fluid flow, wherein the computer for simulating the fluid flow comprises a computer readable medium for storing data associated with a computer program, and a processor for executing a sequence of instructions from the computer program, wherein the computer program is configured to use the fluid model and the model of property balance, and wherein the computer program is operable to run on the computer for simulating the fluid flow to compute the flow of the model gas in every general point of a plurality of non-moving points in the space occupied by the model gas and to display results of computing, thereby reliably predicting the fluid flow including the flow of rarefied gases.
24 . The method of claim 23 , further comprising:
treating each point of a plurality of points in the space occupied by the model gas as a point of collisions for each of the plurality of ballistic particles having the ballistic trajectory with a same ending point; treating each of a plurality of points of collisions as either a point source for each of the plurality of diverging ballistic particles or a point sink for each of the plurality of converging ballistic particles; and treating each of the plurality of ballistic particles moving from the point source to the point sink as a property carrier created in the point source by obtaining one or more of properties of specific values being intrinsic to the model gas surrounding the point source, and ended in the point sink by transferring one or more of properties of specific values in the point sink, wherein a value of property delivered by each of the plurality of ballistic particles converging in the point sink is evaluated regarding whether the value of property is conserved, whether the value of property is changed because of aging, and whether the value of property is modified because of interaction with an external field during a ballistic traveling time.
25 . The method of claim 23 , further comprising:
treating each of a plurality of collisions on a gas-solid interface of the model gas system, which has resulted in diffuse particle scattering from the gas-solid interface, as an act of interaction involving a property transfer from the gas-solid interface to a scattered particle; and treating each of a plurality of points of diffuse particle scattering on the gas-solid interface as a heterogeneous point source for each of a plurality of scattered particles, wherein the gas-solid interface reveals mixed diffuse and specular particle scatterings, wherein velocity of each of a plurality of heterogeneous point sources on the gas-solid interface is assigned to be equal to the velocity of the gas-solid interface in each of a plurality of corresponding points of diffuse particle scattering, and wherein point source strength of each of the plurality of heterogeneous point sources on the gas-solid interface is assigned to be directly proportional to a property accommodation coefficient in each of the plurality of corresponding points of diffuse particle scattering, the property accommodation coefficient which is the probability, for an incident particle, to accommodate one or more of properties intrinsic to the gas-solid interface and to scatter back in the model gas as a diffuse particle, the property accommodation coefficient being in a range from zero to one.
26 . (canceled)
27 . (canceled)
28 . A special purpose computer, comprising:
a processor; a non-transitory computer readable memory coupled to the processor; a user interface coupled to the processor; a display coupled to the processor; and a program stored in the non-transitory computer readable memory and executable by the processor, wherein the program is operable to use a fluid model and a model of property balance, wherein the fluid model comprises treating a fluid flow including a flow of a rarefied gas as a flow of a model gas in a model gas system, the model gas in which particles including molecules are treated as randomly interacting by collisions ballistic particles traveling with a probability between points in a space occupied by the model gas by following a ballistic trajectory governed by a law of motion, wherein the model of property balance in a combination the fluid model transforms parameters characterizing motion of a plurality of ballistic particles into parameters characterizing the fluid flow, and wherein the program is operable when executed to: compute a trajectory and a velocity of a ballistic particle converging in a general non-moving point at a given time; compute a probability of ballistic traveling along a converging ballistic trajectory; compute a net rate of property influx per unit volume in a general non-moving point in a space occupied by the model gas at the given time, which is formed by a flow of converging ballistic particles, each having a preceding collision in the space occupied by the model gas; compute the net rate of property influx per unit volume in the general non-moving point of at the given time, which is formed by flow of converging ballistic particles, each having a preceding scattering from a gas-solid interface; compute the trajectory and the velocity of the ballistic particle diverging from the general non-moving point at the given time; compute the probability of ballistic traveling along a diverging ballistic trajectory; and compute a net rate of property efflux per unit volume, which is formed by diverging ballistic particles from the general non-moving point at the given time, thereby upon formulating a system of mutually dependent integral property balance equations in each of a plurality of non-moving points at the given time and resolving the system of mutually dependent integral property balance equations, obtaining property value in each of the plurality of non-moving points at the given time, and thereby reliably predicting the fluid flow including the flow of the rarefied gas.Join the waitlist — get patent alerts
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