CN113221350A - Hypersonic aircraft transition prediction method based on global stability analysis - Google Patents
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Abstract
The invention relates to a hypersonic aircraft transition prediction method based on global stability analysis, which is realized by the following steps: calculating to obtain a laminar flow basic flow field on the surface of a target high supersonic aircraft to obtain a three-dimensional boundary layer; step two, performing two-dimensional global stability analysis on the three-dimensional boundary layer of the basic flow field obtained in the step one at different flow direction positions to obtain a plurality of growth rate cloud charts of unstable modes; step three, calculating the maximum growth factor of each unstable mode at each flow direction position obtained in the step two, representing the maximum growth factor by using a Nmax curve, and obtaining a Nmax _ all curve by using envelope lines of the Nmax curves of different modes again to represent the conservatively estimated maximum growth factor of the unstable mode; and step four, predicting the transition position of the target hypersonic aircraft according to a transition criterion Nt.
Description
The technical field is as follows:
the invention relates to the technical field of transition prediction of aircrafts, in particular to a transition prediction method of a boundary layer of a hypersonic aircraft, and particularly relates to a transition prediction method of the hypersonic aircraft based on global stability analysis.
Background art:
under the hypersonic flight condition, the friction resistance and the heat flow of the laminar flow area and the turbulent flow area have a great difference, if the transition position can be accurately predicted, when a hypersonic flight vehicle thermal protection system is designed, the problem that the weight of the hypersonic flight vehicle is overlarge due to conservative design completely according to the turbulent flow state can be avoided, so that the flight distance and the maneuverability of the hypersonic flight vehicle are greatly improved, and the hypersonic flight vehicle thermal protection system has important significance for research and development of the hypersonic flight vehicle.
Currently, the frequently adopted transition prediction methods include: the method comprises a traditional eN method based on semi-experience of a linear stability theory, a method (PSE) for solving a parabolic stability equation, a Direct Numerical Simulation (DNS) and a transition mode, wherein the eN method is generally considered to be an effective method for predicting the transition position in engineering application.
The eN method is a transition prediction method based on a linear stability theory, and is traditionally directed at a one-dimensional profile. The transition prediction method predicts the transition by calculating the linear increase multiple of the unstable wave, and the transition criterion, namely the critical value of N, is usually given by depending on experiments or practical experience, so that the transition prediction method is a semi-theoretical semi-empirical transition prediction method.
The amplification factor in the evolution of the amplitude of a frequency disturbance along the flow direction may be e-exponential, i.e. e, of the amplification factor NNExpressed, the expression is:
the expression for the value of the amplification factor N is therefore:
wherein ξ0Is the starting position of integration, xi is any position downstream of the integration path, A0Is an initial position xi0The disturbance amplitude is set, A is the disturbance amplitude at the local xi position, -alphaiIs the spatial growth rate. Calculating N values under different frequencies, and then obtaining envelope curves of the N values under different frequencies, namely NmaxCurve line. For NmaxThe value can be used for proposing a transition criterion Nt according to experiments and practical experience, when N ismaxWhen the value reaches Nt, a transition is considered to occur.
The conventional eN method is directed to a one-dimensional elementary stream profile, and only the variations of the elementary stream profile in the normal direction are considered in the stability analysis. However, the basic flow field on the surface of the hypersonic aircraft mostly has an area which is changed drastically both along the normal direction and the span direction, and a two-dimensional basic flow section is arranged at the same flow direction position, so that the transition position of the area cannot be accurately predicted by the conventional eN method. Therefore, a transition prediction method based on two-dimensional global stability analysis is needed to be provided for the hypersonic aircraft and simultaneously considering the changes of the basic flow profile along the normal direction and the span direction.
The invention content is as follows:
the invention aims to provide a hypersonic aircraft transition prediction method based on global stability analysis, which considers the characteristic that a basic flow section changes violently along the normal direction and the span direction at the same time based on a two-dimensional global stability analysis method, makes up the defect that the conventional eN method only considers the change of the basic flow section along the normal direction, and simultaneously makes N of a plurality of unstable modes changemaxObtaining N by taking envelope again from the curvemax_allAnd (4) obtaining a transition criterion Nt for accurately predicting the transition position, and solving the problem of engineering application of the traditional eN method. The technical solution of the invention is as follows:
a transition prediction method of a hypersonic aircraft based on global stability analysis is realized by the following steps:
calculating to obtain a laminar flow basic flow field on the surface of a target high supersonic aircraft to obtain a three-dimensional boundary layer;
step two, performing two-dimensional global stability analysis on the three-dimensional boundary layer of the basic flow field obtained in the step one at different flow direction positions to obtain a plurality of growth rate cloud charts of unstable modes;
step three, calculating the maximum growth factor of each unstable mode obtained in the step two at each flow direction position, and using NmaxIs shown by a curve and is applied to N of different modesmaxObtaining N by taking envelope again from the curvemax_allA curve representing the maximum growth factor of a conservatively estimated unstable mode;
and step four, predicting the transition position of the target hypersonic aircraft according to a transition criterion Nt.
In the second step, the method for performing two-dimensional global stability analysis at different flow direction positions comprises the following steps: establishing a compressible dimensionless N-S equation in a matrix form by adopting a matrix-free two-dimensional global stability analysis method suitable for a three-dimensional boundary layer of a hypersonic aircraft; expressing the instantaneous quantity as the sum of the basic flow and the disturbance quantity, subtracting the equation of the basic flow from the equation of the instantaneous quantity, and omitting the high-order small quantity to obtain a linear disturbance equation; and then, expressing the disturbance into a flow direction wave form by using a local approximate parallel flow hypothesis, and finally obtaining a frequency-free two-dimensional global stability equation required by matrix-free two-dimensional global stability analysis.
Further, the method for performing two-dimensional global stability analysis at different flow direction positions is performed according to the following steps:
1) establishing a compressible dimensionless N-S equation in a matrix form under a Cartesian coordinate system:
wherein x, y, z respectively represent the flow direction, normal direction and spreading direction in a cartesian coordinate system, phi represents the instantaneous vector phi ═ p, u, v, w, T)TWhere ρ represents density, u represents flow velocity, v represents normal velocity, w represents spanwise velocity, T represents temperature, Γ0、A0、B0、C0、D0、V11、V22、V33、V12、V23、V13Coefficient matrices representing the linear part of the equation, FnRepresenting a non-linear portion;
2) expressing the instantaneous quantity as the sum of the basic flow and the disturbance quantity, i.e.
Wherein
Representing the amount of a steady elementary stream,
the density is expressed as a function of time,
the speed of the flow direction is indicated,
which is indicative of the normal velocity of the wire,
the speed of the span-wise direction is indicated,
represents the temperature; phi '═ of (ρ', u ', v', w ', T')TFor the disturbance amount, ρ ' represents density, u ' represents flow direction velocity, v ' represents normal velocity, w ' represents span direction velocity, and T ' represents temperature; instantaneous quantity phi and basic flow
The equation of the basic flow is subtracted from the equation of the instantaneous quantity, and the nonlinear high-order small quantity is omitted, so that the linear disturbance equation under a Cartesian coordinate system can be obtained, and the form is as follows:
wherein the variable φ 'represents the perturbation vector (ρ', u ', v', w ', T')T，
Γ、A、B、C、D、Vxx、Vyy、Vzz、Vxy、Vyz、VxzCoefficient matrix representing linear disturbance equation, in which basic flow is contained
The information of (a);
3) the linear disturbance equation (2) obtains a linear disturbance equation under a curve coordinate system through coordinate transformation:
where ξ, η, ζ denote the flow direction, normal direction and spread direction respectively in a curved coordinate system, and Φ 'denotes a disturbance vector Φ' ((ρ ', u', v ', w', T))T，Γ、
D、Vξξ、Vηη、
Vξη、
Coefficient matrix representing linear disturbance equation in curve coordinate system, including basic flow
The expression of each coefficient matrix is
Vξξ＝Vxx,
Vξη＝ηyVxy+ηzVxz,
Vξζ＝ζyVzy+ζzVxz,
Vηζ＝2ηyζyVyy+2ηzζzVzz+(ηyζz+ηzζy)Vyz。 (4)
Wherein J represents Jacobi matrix determinant with expression of
4) Deriving a control equation for the matrix-free two-dimensional global stability method, i.e. a frequency-free two-dimensional global stability equation, using a local parallel flow assumption to set the disturbance in the form of a flow direction waveform
Wherein
A function of the shape representing the time t,
wherein
The density is expressed as a function of time,
the speed of the flow direction is indicated,
which is indicative of the normal velocity of the wire,
the speed of the span-wise direction is indicated,
denotes temperature, α denotes flow direction wave number, c.c. denotes complex conjugation; substituting the formula (6) into a linear disturbance equation (3) under a curve coordinate system, and sorting to obtain a control equation used by the matrix-free global stability method, namely a frequency-free two-dimensional global stability equation:
wherein gamma is,
Coefficient matrix representing control equation, in which elementary streams are included
And a real number form flow direction wave number alpha in a time mode, the expression of each coefficient matrix being
Further, in the second step, the method for obtaining the growth rate cloud charts of the plurality of unstable modes comprises: calculating a time mode characteristic value lambda of a two-dimensional global stability equation (7) by adopting an Arnoldi method, and obtaining a disturbed time mode growth rate omega through the relation between the characteristic value lambda and the frequency omegai(ii) a Then gain the growth rate-alpha of the space mode which is more in line with the actual physical situation through Gaster transformationi(αiWhich is the imaginary part of the flow direction wave number α in complex form in spatial mode), the form of the Gaster transform is: - αi＝ωi/cgWherein c isgFor group velocity, the expression is
Finally, the base in the step one can be obtainedGrowth rate-alpha of different flow direction positions of the flow field at various frequenciesiAnd a characteristic function, thereby obtaining a contour cloud picture of the growth rate of a plurality of unstable modes.
Further, in step two, for equation (7), the relationship between the eigenvalue λ and the frequency ω is
Wherein, step III Nmax_allThe curve is obtained as follows: integrating the growth rate of each most unstable mode along the flow direction based on the growth rate cloud pictures of a plurality of unstable modes under different frequencies to obtain the distribution of the amplitude amplification factor N value of each frequency disturbance along the flow direction coordinate, enveloping the maximum value of the N value of different frequency waves along the flow direction coordinate distribution by using a curve, and representing the maximum amplitude amplification factor which can be reached by each frequency disturbance at the mode by using an envelope curve, namely obtaining the NmaxA curve; then, taking the guard estimate, and combining N of different modesmaxThe curve is wrapped again to obtain the maximum amplitude amplification factors of a plurality of unstable modes, namely Nmax_allCurve line.
The determination method of the transition criterion Nt in the fourth step is as follows: the transition criterion Nt is obtained and corrected through flight experiments or ground wind tunnel experiment data, and the amplitude of disturbance with a certain frequency reaches e of the initial amplitude in the process of propagating towards the flow directionNtThe laminar flow develops into turbulent flow; n obtained in step threemax_allFinding the flow direction position corresponding to Nt on the curve, namely considering the transition starting position Xt of the hypersonic aircraft surface.
The invention provides a hypersonic aircraft transition prediction method based on global stability analysis, which has the advantages compared with the prior art that:
(1) the hypersonic aircraft surface has a region with a basic flow field which changes violently along the normal direction and the span direction, and the transition position of the region can be predicted by the method.
(2) The invention adopts a two-dimensional global stability analysis method, considers the changes of the basic flow profile along the normal direction and the span direction, has physical basis and has more reliable transition prediction result.
Description of the drawings:
FIG. 1 is a block diagram of the steps of the present invention
FIG. 2 is a schematic diagram of a model of a target hypersonic aircraft
FIG. 3 is a cloud chart of elementary stream flow direction velocity of a target hypersonic aircraft model
FIG. 4 is a cloud chart of growth rates of 6 most unstable modes in boundary layers of basic flow fields of an aircraft model
FIG. 5 shows N in the 6 most unstable modesmaxCurve
FIG. 6 shows N in the 6 most unstable modesmax_allCurve
The specific implementation mode is as follows:
the present invention will be described in detail with reference to the following embodiments and accompanying drawings. In the following description, for purposes of explanation and not limitation, specific details are set forth in order to provide a thorough understanding of the present invention. However, it will be apparent to one skilled in the art that the present invention may be practiced in other embodiments that depart from these specific details.
It should be noted that, in order to avoid obscuring the present invention with unnecessary details, only the device structures and/or processing steps that are closely related to the scheme according to the present invention are shown in the drawings, and other details that are not so relevant to the present invention are omitted.
The embodiment of the invention provides a hypersonic aircraft transition prediction method based on global stability analysis, which comprises the following steps of:
step one, calculating to obtain a laminar flow basic flow field of a target high supersonic aircraft;
in this step, the target hypersonic aircraft of this embodiment is an elliptical cone model with a ratio of major axis to minor axis of 3:1, the radius of the blunt tip is 0.95mm, the radius of the minor axis is 41mm, the flight mach number Ma is 6, the angle of attack is 0 degree, the sideslip angle is 0 degree, a basic flow field is calculated by Direct Numerical Simulation (DNS), the characteristic length used without quantization is the radius of the blunt tip, the characteristic speed is the free incoming flow speed, and the aircraft model is schematically shown in fig. 2;
in the step, the number of normal grids in the boundary layer is required to be more than 100 so as to meet the requirement of subsequent global stability analysis; the calculated flow direction velocity cloud chart of the elementary flow field is shown in fig. 3, and only the result of the 1/4 model is calculated due to symmetry;
step two, performing two-dimensional global stability analysis on the three-dimensional boundary layer of the basic flow field obtained in the step one at different flow direction positions to obtain a plurality of growth rate cloud charts of unstable modes;
in this step, the global stability analysis step is as follows:
1) starting from a compressible dimensionless N-S equation in matrix form in a cartesian coordinate system, the equation is as follows:
wherein x, y, z respectively represent the flow direction, normal direction and spreading direction in a cartesian coordinate system, phi represents the instantaneous vector phi ═ p, u, v, w, T)TWhere ρ represents density, u represents flow velocity, v represents normal velocity, w represents spanwise velocity, T represents temperature, Γ0、A0、B0、C0、D0、V11、V22、V33、V12、V23、V13Coefficient matrices representing the linear part of the equation, FnRepresenting the nonlinear part, the matrix elements are specifically referred to: transition mechanism and prediction of peristatic, Su rainbow, Zhang Yongming, supersonic/hypersonic boundary layer [ M]Beijing, science publishers;
2) expressing the instantaneous quantity as the sum of the basic flow and the disturbance quantity, i.e.
Wherein
Representing the amount of a steady elementary stream,
the density is expressed as a function of time,
the speed of the flow direction is indicated,
which is indicative of the normal velocity of the wire,
the speed of the span-wise direction is indicated,
represents the temperature; phi '═ of (ρ', u ', v', w ', T')TFor the disturbance amount, ρ ' represents density, u ' represents flow direction velocity, v ' represents normal velocity, w ' represents span direction velocity, and T ' represents temperature. Instantaneous quantity phi and basic flow
The equation of the basic flow is subtracted from the equation of the instantaneous quantity, and the nonlinear high-order small quantity is omitted, so that the linear disturbance equation under a Cartesian coordinate system can be obtained, and the form is as follows:
wherein the variable phi' represents a disturbance vector
Γ、A、B、C、D、Vxx、Vyy、Vzz、Vxy、Vyz、VxzCoefficient matrix representing linear disturbance equation, in which basic flow is contained
The coefficient matrix elements refer to: stability characteristics and disturbance evolution of Zhang Shaolong-hypersonic 2:1 elliptic cone boundary layer [ D]Tianjin university, 2016;
3) equation (2) obtains a linear disturbance equation under a curve coordinate system through coordinate transformation
Where ξ, η, ζ denote the flow direction, normal direction and spread direction respectively in a curved coordinate system, and Φ 'denotes a disturbance vector Φ' ((ρ ', u', v ', w', T))T，Γ、
D、Vξξ、Vηη、
Vξη、
Coefficient matrix representing linear disturbance equation in curve coordinate system, including basic flow
The expression of each coefficient matrix is
Vξξ＝Vxx,
Vξη＝ηyVxy+ηzVxz,
Vξζ＝ζyVzy+ζzVxz,
Vηζ＝2ηyζyVyy+2ηzζzVzz+(ηyζz+ηzζy)Vyz。 (4)
Wherein J represents Jacobi matrix determinant with expression of
4) A control equation used by a matrix-free two-dimensional global stability method, namely a frequency-free two-dimensional global stability equation, is obtained through derivation, and the time mode problem is solved. Using the assumption of local parallel flow, the perturbation is set to flow direction wave form
Wherein
A function of the shape representing the time t,
wherein
The density is expressed as a function of time,
the speed of the flow direction is indicated,
which is indicative of the normal velocity of the wire,
the speed of the span-wise direction is indicated,
denotes temperature, α denotes flow direction wave number, and c.c. denotes complex conjugation. The concepts and methods referred to herein are well known in the art, and reference is made specifically to: zhongheng, Zhao Tuff, flow stability [ M]National defense industry publishers, 2004.
Substituting the formula (6) into a linear disturbance equation (3) under a curve coordinate system, and sorting to obtain a control equation used by the matrix-free global stability method, namely a frequency-free two-dimensional global stability equation
Wherein gamma is,
Coefficient matrix representing control equation, in which elementary streams are included
And a real number form flow direction wave number alpha in a time mode, the expression of each coefficient matrix being
5) Calculating a time mode characteristic value lambda of a two-dimensional global stability equation (7) by adopting an Arnoldi method, and obtaining a disturbed time mode growth rate omega through the relation between the characteristic value lambda and the frequency omegai(ii) a Then gain the growth rate-alpha of the space mode which is more in line with the actual physical situation through Gaster transformationi(αiIs in the form of complex number in space modeTowards the imaginary part of the wave number α); finally, the growth rate-alpha of different flow direction positions of the basic flow field in the step one at each frequency can be obtainediAnd a characteristic function, thereby obtaining a contour cloud picture of growth rates of a plurality of unstable modes;
in this step, the Arnoldi method is a method known in the art, and can be implemented by calling an open source ARPACK package, specifically referring to: R.B.Lehoucq, D.C.Sorensen, C.Yang.ARPACK Users Guide Solution of Large Scale Eigenvalue schemes by Implicitly corrected Arnoldi Methods [ J ]. Journal of Chemical Physics,2015,157(S14): 044119.
In this step, the relationship between the characteristic value λ and the frequency ω is
In particular, equation (7) relates to
Defines a linear evolution operator L (t, α) for a certain wavenumber α, which can be expressed as
Taking a fixed time interval T0Then there is
Wherein L (T)0α) is still a linear operator, then it has a characteristic modality, i.e.
Wherein the value of the lambda is a characteristic value,
is a characteristic function corresponding to λ. On the other handIn equation (7), corresponding to a characteristic mode
From T-0 to T-T0Can also be described in complex form as frequency ω
Where the real part of ωrRepresenting frequency, imaginary part omegaiIndicating the rate of growth. For the characteristic modality, it can be described as
In summary, the relationship between the eigenvalue λ and the frequency ω is
In this step, the Gaster transform is in the form:
-αi＝ωi/cg。 (14)
wherein c isgFor group velocity, the expression is
This method is well known in the art, with specific reference to: gate M.A note on the relationship between the temporal interaction and the spatial interaction distribution in the dynamic status [ J].Journal of Fluid Mechanics,1962,14(2):222-224。
In the step, for a target hypersonic aircraft model, cloud charts of growth rates of 6 most unstable modes distributed along the flow direction are obtained, and the cloud charts are respectively Y-Mode1, Y-Mode2, Y-Mode3, Z-Mode1, Z-Mode2 and Z-Mode3, and are shown in FIG. 4;
step three, calculating the maximum growth factor of each unstable mode obtained in the step two at each flow direction position, and using NmaxCurveIs shown and envelope is again taken to obtain Nmax_allA curve representing the maximum growth factor of a conservatively estimated unstable mode;
in this step, it is considered that the perturbations of different frequencies in the boundary layer can continuously increase to the transition position, and the amplification factor of the amplitude of the small perturbation of a given frequency from the initial position to a certain position is given by equation (14):
in which ξ0Is the starting position of integration, xi is any position downstream of the integration path, A0Is an initial position xi0The disturbance amplitude is set, A is the disturbance amplitude at the local xi position, -alphaiIs the spatial growth rate.
In the step, based on a plurality of growth rate cloud pictures of unstable modes, the growth rate of the most unstable mode is integrated along the flow direction under different frequencies, the distribution of amplitude amplification factor N values of various frequency disturbances along the flow direction coordinate can be obtained, the maximum values of the N values of different frequency waves along the flow direction coordinate distribution are enveloped by a curve, the envelope represents the maximum amplitude amplification factor which can be reached by various frequency disturbances, and the N is obtainedmaxA curve; n of the 6 most unstable modes of the present embodimentmaxThe curves are shown in FIG. 5; to take a conservative estimate, the N ismaxThe curve is wrapped again to obtain the maximum amplitude amplification factors of a plurality of unstable modes, namely Nmax_alCurve l, N of this examplemax_allThe curves are shown in FIG. 6;
predicting the transition position of the surface of the target hypersonic aircraft according to a transition criterion Nt;
in this step, it is considered that the amplitude reaches e of the initial amplitude as long as the disturbance of a certain frequency is in the process of propagating to the flow directionNtThe laminar flow develops into turbulent flow; n obtained in step threemax_allFinding a flow direction position corresponding to Nt on the curve, namely considering the transition starting position Xt of the hypersonic aircraft surface;
in this step, the transition criterion Nt of this embodiment is 8.5, and is obtained by calibrating a wind tunnel experiment result and a numerical calculation result of a known hypersonic aircraft model.
In this step, N of the target hypersonic aircraft model is combined in this embodimentmax_allThe transition position of the 3:1 elliptical cone short axis region is predicted to be about X to 247 by the curve (fig. 6) and the transition criterion Nt to 8.5.
Features that are described and/or illustrated above with respect to one embodiment may be used in the same way or in a similar way in one or more other embodiments and/or in combination with or instead of the features of the other embodiments.
It should be emphasized that the term "comprises/comprising" when used herein, is taken to specify the presence of stated features, integers, steps or components but does not preclude the presence or addition of one or more other features, integers, steps, components or groups thereof.
The above devices and methods of the present invention can be implemented by hardware, or can be implemented by hardware and software. The present invention relates to a computer-readable program which, when executed by a logic section, enables the logic section to realize the above-described apparatus or constituent section, or to realize the above-described various methods or steps. The present invention also relates to a storage medium such as a hard disk, a magnetic disk, an optical disk, a DVD, a flash memory, or the like, for storing the above program.
The many features and advantages of these embodiments are apparent from the detailed specification, and thus, it is intended by the appended claims to cover all such features and advantages of these embodiments which fall within the true spirit and scope thereof. Further, since numerous modifications and changes will readily occur to those skilled in the art, it is not desired to limit the embodiments of the invention to the exact construction and operation illustrated and described, and accordingly, all suitable modifications and equivalents may be resorted to, falling within the scope thereof.
The invention has not been described in detail and is in part known to those of skill in the art.
Claims (6)
1. A transition prediction method of a hypersonic aircraft based on global stability analysis is realized by the following steps:
step one, calculating to obtain a laminar flow basic flow field on the surface of the target hypersonic aircraft, and obtaining a three-dimensional boundary layer.
Step two, performing two-dimensional global stability analysis on the three-dimensional boundary layer of the basic flow field obtained in the step one at different flow direction positions to obtain a plurality of growth rate cloud charts of unstable modes, wherein the method for performing the two-dimensional global stability analysis at different flow direction positions comprises the following steps: establishing a compressible dimensionless N-S equation in a matrix form by adopting a matrix-free two-dimensional global stability analysis method suitable for a three-dimensional boundary layer of a hypersonic aircraft; expressing the instantaneous quantity as the sum of the basic flow and the disturbance quantity, subtracting the equation of the basic flow from the equation of the instantaneous quantity, and omitting the high-order small quantity to obtain a linear disturbance equation; then, using a local approximate parallel flow hypothesis to express the disturbance into a flow direction wave form, and finally obtaining a two-dimensional global stability equation without frequency required by the matrix-free two-dimensional global stability analysis;
step three, calculating the maximum growth factor of each unstable mode obtained in the step two at each flow direction position, and using NmaxIs shown by a curve and is applied to N of different modesmaxObtaining N by taking envelope again from the curvemax_allA curve representing the maximum growth factor of a conservatively estimated unstable mode;
and step four, predicting the transition position of the target hypersonic aircraft according to a transition criterion Nt.
2. The method for predicting transition of the hypersonic aircraft based on global stability analysis according to claim 1, wherein the method for performing two-dimensional global stability analysis at different flow direction positions is performed according to the following steps:
1) establishing a compressible dimensionless N-S equation in a matrix form under a Cartesian coordinate system:
1)
wherein x, y, z respectively represent the flow direction, normal direction and spreading direction in a cartesian coordinate system, phi represents the instantaneous vector phi ═ p, u, v, w, T)TWhere ρ represents density, u represents flow velocity, v represents normal velocity, w represents spanwise velocity, T represents temperature, Γ0、A0、B0、C0、D0、V11、V22、V33、V12、V23、V13Coefficient matrices representing the linear part of the equation, FnRepresenting a non-linear portion;
2) expressing the instantaneous quantity as the sum of the basic flow and the disturbance quantity, i.e.
Wherein
Representing the amount of a steady elementary stream,
the density is expressed as a function of time,
the speed of the flow direction is indicated,
which is indicative of the normal velocity of the wire,
the speed of the span-wise direction is indicated,
represents the temperature; phi '═ of (ρ', u ', v', w ', T')TFor the disturbance amount, ρ ' represents density, u ' represents flow direction velocity, v ' represents normal velocity, w ' represents span direction velocity, and T ' represents temperature; instantaneous quantity phi and basic flow
The equation of the basic flow is subtracted from the equation of the instantaneous quantity, and the nonlinear high-order small quantity is omitted, so that the linear disturbance equation under a Cartesian coordinate system can be obtained, and the form is as follows:
2)
wherein the variable φ 'represents the perturbation vector (ρ', u ', v', w ', T')T，
Γ、A、B、C、D、Vxx、Vyy、Vzz、Vxy、Vyz、VxzCoefficient matrix representing linear disturbance equation, in which basic flow is contained
The information of (a);
3) the linear disturbance equation (2) obtains a linear disturbance equation under a curve coordinate system through coordinate transformation:
3)
where ξ, η, ζ denote the flow direction, normal direction and spread direction respectively in a curved coordinate system, and Φ 'denotes a disturbance vector Φ' ((ρ ', u', v ', w', T))T，Γ、
D、Vξξ、Vηη、
Vξη、
Coefficient matrix representing linear disturbance equation in curve coordinate system, including basic flow
The expression of each coefficient matrix is
4)Vξξ＝Vxx,
Vξη＝ηyVxy+ηzVxz,
Vξζ＝ζyVzy+ζzVxz,
Vηζ＝2ηyζyVyy+2ηzζzVzz+(ηyζz+ηzζy)Vyz。 (4)
Wherein J represents Jacobi matrix determinant with expression of
5)
4) Deriving a control equation for the matrix-free two-dimensional global stability method, i.e. a frequency-free two-dimensional global stability equation, using a local parallel flow assumption to set the disturbance in the form of a flow direction waveform
6)
Wherein
A function of the shape representing the time t,
wherein
The density is expressed as a function of time,
the speed of the flow direction is indicated,
which is indicative of the normal velocity of the wire,
the speed of the span-wise direction is indicated,
denotes temperature, α denotes flow direction wave number, c.c. denotes complex conjugation; substituting the formula (6) into a linear disturbance equation (3) under a curve coordinate system, and sorting to obtain a control equation used by the matrix-free global stability method, namely a frequency-free two-dimensional global stability equation:
7)
wherein gamma is,
Coefficient matrix representing control equation, in which elementary streams are included
And a real number form flow direction wave number alpha in a time mode, the expression of each coefficient matrix being
8)
3. The method for predicting transition of hypersonic aircraft according to claim 1,
the method for obtaining the growth rate cloud pictures of the multiple unstable modes in the second step is characterized in that the method for obtaining the growth rate cloud pictures of the multiple unstable modes comprises the following steps: calculating a time mode characteristic value lambda of a two-dimensional global stability equation (7) by adopting an Arnoldi method, and obtaining a disturbed time mode growth rate omega through the relation between the characteristic value lambda and the frequency omegai(ii) a Then gain the growth rate-alpha of the space mode which is more in line with the actual physical situation through Gaster transformationi，αiIn the form of complex flow in the spatial mode, towards the imaginary part of the wave number α, the Gaster transform is of the form: - αi＝ωi/cgWherein c isgFor group velocity, the expression is
Finally, the growth rate-alpha of different flow direction positions of the basic flow field in the step one at each frequency can be obtainediAnd a characteristic function, thereby obtaining a contour cloud picture of the growth rate of a plurality of unstable modes.
4. The method for predicting transition of hypersonic aircraft according to claim 3, wherein in the second step, for equation (7), the relation between the eigenvalue λ and the frequency ω is
5. The method for predicting transition of hypersonic aircraft according to claim 1, wherein the step three N is a stepmax_allThe curve is obtained as follows: integrating the growth rate of each most unstable mode along the flow direction based on the growth rate cloud pictures of a plurality of unstable modes under different frequencies to obtain the distribution of the amplitude amplification factor N value of each frequency disturbance along the flow direction coordinate, enveloping the maximum value of the N value of different frequency waves along the flow direction coordinate distribution by using a curve, and representing the maximum amplitude amplification factor which can be reached by each frequency disturbance at the mode by using an envelope curve, namely obtaining the NmaxA curve; then, taking the guard estimate, and combining N of different modesmaxThe curve is wrapped again to obtain the maximum amplitude amplification factors of a plurality of unstable modes, namely Nmax_allCurve line.
6. The method for predicting transition of the hypersonic aircraft based on global stability analysis according to claim 1, wherein the determination method of the transition criterion Nt in the step four is as follows: the transition criterion Nt is obtained and corrected through flight experiments or ground wind tunnel experiment data, and the amplitude of disturbance with a certain frequency reaches e of the initial amplitude in the process of propagating towards the flow directionNtThe laminar flow develops into turbulent flow; n obtained in step threemax_allFinding the flow direction position corresponding to Nt on the curve, namely considering the transition starting position Xt of the hypersonic aircraft surface.
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