A method for obtaining uncertainty of propeller slipstream effect based on arbitrary polynomial chaos expansion

Abstract

A method for obtaining the uncertainty of propeller slipstream effect based on chaotic expansion of arbitrary polynomials. First, statistics are collected on the moments of uncertainty variables during the flight of a propeller aircraft; bases of polynomials of various orders are constructed, and Latin hypercube sampling is performed on the uncertainty variables in the flight data; the system response at the corresponding sampling point is obtained according to CFD simulation; the polynomial coefficients are calculated by the least squares method to obtain the polynomial function of the system response; it is determined whether the highest order of the polynomial is sufficient and whether the goodness of fit of the chaotic polynomial to the system meets the requirements; the flight data collection points corresponding to the uncertainty variables are used as input to obtain the system random response surface; finally, the probability distribution of the system response caused by the uncertainty variables is obtained by mathematical statistics methods. The present invention quantitatively estimates the impact of the uncertainty of the flight parameters of the propeller aircraft on the slipstream, and further, the slipstream uncertainty distribution can still be obtained when the probability distribution of the flight parameters is unknown.

Description

Technical Field

[0001] The present invention relates to the field of propeller effect uncertainty analysis, and in particular to a method for obtaining the uncertainty of propeller slipstream effect based on arbitrary polynomial chaos expansion. Background Art

[0002] Propeller aircraft have been around for more than 100 years. They are widely used in flight training, long-duration cruising and special operations due to their simple operation, good take-off and landing performance, long air control time and low flight cost. They have an irreplaceable position in the field of transport aircraft. However, propeller aircraft will inevitably be affected by slipstream. Propeller slipstream is a non-pipeline controlled flow with complex flow field characteristics. The parts swept by the slipstream, especially the wings and tail, will be strongly disturbed, thus affecting the overall performance of the aircraft. Therefore, studying propeller slipstream is very necessary for modern aircraft design.

[0003] The existing analysis of propeller aerodynamic characteristics mostly adopts the following three methods: theoretical method, experimental method and numerical simulation method. Although the theoretical method can quickly calculate the aerodynamic characteristics of the propeller with high accuracy, it is difficult to solve the problem of propeller slipstream of the whole aircraft with complex structure required in engineering practice, and simple theoretical calculation is difficult to meet engineering needs. As for the experimental method, its result is the closest to the actual working condition, but the R&D cost is too high and the design cycle is long, so it is difficult to be used as the main method for modern aircraft design. Therefore, most studies have adopted the method of numerical simulation to improve flight performance and reduce R&D cost and design cycle. These methods can obtain the influence of propeller slipstream effect on flight performance to some extent, but the size of propeller slipstream effect is affected by many factors, including the structure of the propeller itself, propeller speed and flight angle of attack, etc. These factors are constantly changing during actual flight, and the influence of the uncertainty of each factor on the slipstream effect is difficult to quantify. To this end, many scholars have also proposed the method of chaotic polynomial expansion to quantify the impact of the uncertainty of input variables on the system, including embedded polynomial chaos and non-embedded polynomial chaos methods. These methods have one thing in common, that is, they require the probability distribution of known input variables, and the probability distribution needs to satisfy Askey's rule, which seriously limits the scope of application of the polynomial chaos method. Summary of the invention

[0004] In view of the above problems, the present invention, based on the existing numerical simulation foundation and combined with the polynomial chaos expansion technology, proposes a method for obtaining the uncertainty of propeller slipstream effect based on arbitrary polynomial chaos expansion. This method can study the influence of multiple uncertainty factors on the propeller slipstream effect. Furthermore, the present invention can still obtain the slipstream uncertainty distribution when the probability distribution of flight parameters is unknown, which provides an important theoretical basis for the driving and design of propeller aircraft.

[0005] In order to achieve this purpose, the technical solution adopted by the present invention is:

[0006] A method for obtaining the uncertainty of propeller slipstream effect based on arbitrary polynomial chaos expansion, characterized by comprising the following steps:

[0007] Step 1, statistically analyzing the moments of uncertainty variables during the flight of the propeller aircraft;

[0008] Step 2, construct the basis of polynomials of various orders and determine the moment order;

[0009] Step 3, use Latin hypercube sampling method to sample the flight data collection points, and obtain the system response at the corresponding sampling points according to CFD simulation;

[0010] Step 4, calculating the polynomial coefficients by the least square method to obtain the polynomial function of the system response;

[0011] Step 5, determine whether the highest order of the polynomial is sufficient and whether the goodness of fit of the chaotic polynomial to the system meets the requirements;

[0012] Step 6, obtain the system random response surface according to the output results of the uncertainty variables, and obtain the probability distribution of the system response through mathematical statistics;

[0013] As an improvement, according to the selected uncertainty variables x1, x2, …, x m The flight phases where each uncertainty variable is stable are selected to record the flight data, and then the first 2n-1 order moments of m random variables are counted, where n is the highest order of the polynomial. Where ρ(x) represents the probability distribution function, x i is the i-th sampling point of the random variable x, and N is the number of sampling points of the random variable x.

[0014] As an improvement, the specific steps of step 2 are as follows:

[0015] (1) Using the moments of random variables, we construct a set of orthogonal bases in any given probability space. The k-th order polynomial basis of the random variable x is in YesP (k)The coefficients of (x), according to the properties of the orthogonal basis, any two basis functions satisfy: Among them, P (k) (x) and P (j) (x) are the kth and jth order polynomial bases, δ kj is a non-zero constant, and the sequence number k, j = 0, 1...;

[0016] (2) Normalize the leading coefficients of all polynomial bases and construct a k-order polynomial base P (k) (x) and all polynomial bases with order less than k, the constraints must satisfy:

[0017]

[0018] According to the definition of moment, the above formula can be reduced to:

[0019] Written in matrix form: That is, the first 2k-1 moments are required to construct the first k-order polynomial basis.

[0020] As an improvement, the specific steps of step 3 are as follows:

[0021] (1) Using Latin hypercube sampling method, select a number of random variables from the recorded flight data. The number of random variables N is PC =(m+n)! / (m!n!);

[0022] (2) N PC The sampling points are used as the input conditions for CFD simulation calculations, and the deterministic response results of each sampling point are calculated. i=0,1,…,n-1.

[0023] As an improvement, the specific steps of step 4 are as follows: the system response can be expanded using a chaotic polynomial: where c0, c1, ..., c n-1 is the unknown coefficient, which can be obtained by the least squares method: The polynomial basis P (k) (x) is obtained from step 2, the system response of each sampling point Obtained from step 3. Therefore, the unknown coefficient c of the chaotic polynomial k Determine, thereby determining the chaotic polynomial f(x) of the entire system response.

[0024] As an improvement, the specific steps of step 5 are as follows:

[0025] (1) Compare the n-order and n+1-order results to determine whether the n-order result is close to the n+1-order result. If so, the accuracy of the n-order result is satisfied. If the two results differ greatly, the order needs to be increased, and then the n+1-order and n+2-order results are compared. This process is repeated until the appropriate polynomial order is found.

[0026] (2) After the order is determined, calculate N PC The correlation index between the configuration points and the chaotic polynomial:

[0027] in is the residual sum of squares, is the total deviation square sum, judging the correlation coefficient R 2 Is it close to 1? If so, it means that the chaotic polynomial meets the requirements for the goodness of fit of the system. Otherwise, the number of sampling points N needs to be increased.

[0028] As an improvement, the specific steps of step 6 are as follows: Substitute the N sampling points of the uncertainty variables in the propeller aircraft flight data into the chaotic polynomial of the system response Get the system response f(x) at each sampling point i ), i = 1, 2, ..., N, and statistically analyze the N system responses to obtain the probability distribution of the system responses caused by the uncertainty variables.

[0029] Beneficial effects:

[0030] Compared with the prior art, the method for obtaining the uncertainty of propeller slipstream effect based on arbitrary polynomial chaos expansion of the present invention can quantitatively analyze the influence of uncertainty variables on slipstream during the flight of propeller aircraft; compared with traditional sampling methods, the present invention requires fewer probability distribution points for the same accuracy; compared with traditional chaotic polynomial expansion methods, the present invention can calculate problems related to the distribution of arbitrary random variables and has wider applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 is a flow chart of an embodiment of the present invention;

[0032] Figure 2 Schematic diagram of the main components of an aircraft used in an actual flight in an embodiment of the method of the present invention, wherein 1-propeller, 2-wing, 3-fuselage, 4-horizontal tail, 5-vertical tail;

[0033] Figure 3 is a change in angle of attack over time and a histogram in a flight test of an embodiment of the present invention;

[0034] Figure 4 3. It is a comparison diagram of the 3rd order and 4th order calculated flow fields of the embodiment of the present invention;

[0035] Figure 5 is the mean and standard deviation distribution of the pressure coefficient of the embodiment of the present invention;

[0036] Figure 6 It is a propagation cloud diagram of slipstream uncertainty in the flow field caused by the angle of attack of an embodiment of the present invention. DETAILED DESCRIPTION

[0037] The following is a detailed description of the technical solution of the present invention in conjunction with a propeller aircraft actual flight experiment. Figure 2 As shown in Figure 1, the main components that affect aerodynamic performance include 1-propeller, 2-wing, 3-fuselage, 4-horizontal tail, and 5-vertical tail. Its main performance parameters and structural parameters are shown in Table 1.

[0038] Table 1 Main structural parameters and performance parameters of propeller aircraft

[0039] length 8.28m high 2.72m span 11.0m Cabin altitude 1.22m airfoil NACA 2412 Cabin width 1.73m Wing area <![CDATA[16.2m 2 ]]> engine IO-360-L2A Maximum airspeed limit 302km / h Top speed 228km / h voyage 1289km Practical ceiling 4100m Climb rate 3.66m / s Standard weight of aircraft 744kg Maximum take-off and landing weight 744kg Maximum load 371kg Luggage allowed 54kg Fuel tank 212L Takeoff ground roll 288.04m Landing ground run 167.64m

[0040] During the flight, the aircraft flew horizontally for 100 knots with an average altitude of 863m. The sensor obtained that the ambient temperature was 29°C and the pressure was 29.97Pa. The change of the angle of attack over time during horizontal flight is shown in the figure below. Figure 3 shown.

[0041] It can be seen that the angle of attack fluctuates in the range of -0.7° to 2.4° during the flight. From the histogram, it can be seen that the angle of attack has obvious uncertainty. The Kolmogorov–Smirnov method is used to perform a normality test on the four continuous random variable distribution models in the Askey criterion. The fitting results are shown in Table 2.

[0042] Table 2 Kolmogorov–Smirnov test fit

[0043] Distribution Model Statistics Goodness of fit ranking Beta 0.05605 1 Gamma 0.05662 2 Normal 0.06411 3 Uniform 0.0901 4

[0044] It can be seen from the table that the uniform distribution has the worst fitting effect, while the other three distributions are similar. The sample cumulative distribution function obtained by the best fitting Beta distribution also differs from the theoretical distribution function by 0.05605 (the closer to 0, the better the effect). Therefore, the samples of this embodiment are poorly fitted by the four continuous random variable distribution models in the Askey criterion, resulting in a large error based on the traditional chaotic polynomial expansion method. For this embodiment, it is difficult to quantify the impact of angle of attack uncertainty on slipstream.

[0045] The arbitrary polynomial chaos method used in the present invention does not require the angle of attack to have a known probability distribution. It only needs to count the first 2n-1 moments of the angle of attack, where n is the polynomial order. Generally, the third-order accuracy is sufficient. This embodiment only studies the influence of the uncertainty of the angle of attack on the slipstream. The calculation conditions of the deterministic flow field are determined according to the actual flight conditions. The slipstream influence is represented by the difference between the propeller rotating and stationary states. In order to more intuitively display the influence of the slipstream on the wing, a cross section is made at x=0.1m parallel to the symmetry plane of the aircraft, and the pressure coefficient C of the point on the cross section is used. p As a deterministic calculation result.

[0046] According to the method flow of the present invention, the polynomial basis and polynomial coefficients are constructed to obtain a chaotic polynomial model that can reflect the system response. Figure 4 As shown, the flow field shown in the figure is the difference between the pressure coefficient calculated under 4th order accuracy and 3rd order accuracy.

[0047] From the figure, we can see that the maximum deviation of the pressure coefficient under the two accuracies is 2%, and the deviation is mainly concentrated at the leading edge of the propeller aircraft wing. It can be seen that the third-order accuracy and the fourth-order accuracy results are not much different, and the third-order accuracy results can meet the requirements. For the third-order chaotic polynomial, its correlation coefficient R 2 The calculated value is 0.99964, which is close to 1. It is considered that the goodness of fit meets the requirements, that is, the obtained chaotic polynomial can be used as a system response model.

[0048] In order to obtain the effect of slipstream, we used commercial CFD software to simulate the pressure coefficient on the wing surface of a propeller aircraft under two conditions (with and without propeller). Figure 5 As shown, it is the pressure coefficient C on the slice of the wing at x = -1m p The airflow flows along the negative direction of the Z axis, and the positions of the leading and trailing edges are as follows: Figure 5 The mean and standard deviation values ​​are expressed in error bands, the midpoint of the error band is the mean, and the upper and lower edges of the error band are the standard deviation range. The gray error band is the result without propellers, and the black error band is the result with propellers. The pressure coefficient C between the two conditions p The mean values ​​are not much different because the C of the upper and lower surfaces of the wing at high airspeed p The absolute value increases sharply, and the C p The absolute value changes are relatively small. In addition, the larger standard deviations are mainly concentrated on the leading edge of the wing. This is mainly because the angle of attack of the flight test is small and the impact of the incoming flow on the upper and lower surfaces of the wing leading edge is similar. In addition, since the leading edge is the windward surface, the airflow separates up and down, resulting in increased leading edge uncertainty. At the same time, the slipstream reduces the standard deviation of the leading edge of the lower wing, and the extreme value area of ​​the standard deviation of the upper wing moves backward from the leading edge, close to 20% of the chord length.

[0049] The standard deviation of the statistical chaos polynomial at N sampling points is the slipstream uncertainty change caused by the attack angle uncertainty. In this embodiment, N is 1227 sampling points, which is mainly determined by the sensor acquisition frequency. Figure 6 The figure shows the pressure coefficient C on the cross section at x = 0.1m. p Standard deviation distribution cloud diagram, where Figure 6 (a) represents the result without propeller. Figure 6 (b) shows the result under the condition of propeller. The difference between the two simulation results is used to characterize the flow field change caused by slipstream. By comparing the results under the two conditions, the following conclusions can be drawn:

[0050] (1) The slipstream standard deviation caused by the angle of attack uncertainty is unevenly distributed in the flow field. The slipstream standard deviation around the wing is larger than that around the tail, that is, the uncertainty is larger, and the maximum value exceeds 0.1. In the flow field around the wing, the slipstream uncertainty is mainly concentrated on the upper and lower wing surfaces at 20% of the chord length of the wing leading edge, 70% of the chord length, and the wake area, that is, the extreme value area of ​​slipstream uncertainty corresponds to the separation area of ​​the airflow. In addition, the slipstream reduces the overall standard deviation of the flow field near the wing and moves the extreme value area toward the trailing edge.

[0051] (2) For the tail, the slipstream uncertainty mainly occurs at the leading edge and the trailing edge. In addition, the propeller slipstream has a downwash effect on the wake, and the horizontal tail is just in the downwash flow field of the slipstream, which causes the local angle of attack of the horizontal tail to decrease. Figure 6 In the flight test shown in (a), there is no slipstream effect, the horizontal tail angle of attack is close to 0°, and there are extreme standard deviation areas on the upper and lower surfaces of the leading edge. Figure 6 (b) The downwash effect of the slipstream causes the horizontal tail to form a negative angle of attack, so the standard deviation produces only an extreme value area on the lower surface of the leading edge.

[0052] The above are only specific steps of the present invention and do not constitute any limitation to the protection scope of the present invention; any technical solutions formed by equivalent transformation or equivalent replacement shall fall within the protection scope of the present invention; some parts not elaborated in detail in the present invention belong to the known technologies of those skilled in the art.

Claims

1. A method for obtaining the uncertainty of propeller slipstream effect based on arbitrary polynomial chaos expansion, characterized in that: The following steps are involved: Step 1, statistically analyzing the moments of uncertainty variables during the flight of the propeller aircraft; Step 2, construct the basis of polynomials of various orders and determine the moment order; Step 3, use Latin hypercube sampling method to sample the flight data collection points, and obtain the system response at the corresponding sampling points according to CFD simulation; Step 4, calculating the polynomial coefficients by the least square method to obtain the polynomial function of the system response; Step 5, determine whether the highest order of the polynomial is sufficient and whether the goodness of fit of the chaotic polynomial to the system meets the requirements; Step 6, obtain the system random response surface according to the output results of the uncertainty variables, and obtain the probability distribution of the system response through mathematical statistics; The specific steps of step 1 are as follows: According to the selected uncertainty variables x1, x2, ..., x m Select the flight phase where each uncertainty variable is stable to record the flight data, and then count the first 2n-1 order moments of m random variables, where n is the highest order of the polynomial, and the kth order moment of the random variable x is Where ρ(x) represents the probability distribution function, x i is the i-th sampling point of the random variable x, and N is the number of sampling points of the random variable x; The specific steps of step 2 are as follows: (1) Using the moments of random variables, we construct a set of orthogonal bases in any given probability space. The k-th order polynomial basis of the random variable x is in YesP (k) The coefficients of (x), according to the properties of the orthogonal basis, any two basis functions satisfy: Among them, P (k) (x) and P (j) (x) are the kth and jth order polynomial bases, δ kj is a non-zero constant, and the sequence number k, j = 0, 1...; (2) Normalize the leading coefficients of all polynomial bases and construct a k-order polynomial base P (k) (x) and all polynomial bases with order less than k, the constraints must satisfy: According to the definition of moment, the above formula can be reduced to: Written in matrix form: That is, the first 2k-1 moments are required to construct the first k-order polynomial basis.

2. The method for obtaining the uncertainty of propeller slipstream effect based on arbitrary polynomial chaos expansion according to claim 1 is characterized in that: The specific steps of step 3 are as follows: (1) Using Latin hypercube sampling method, select a number of random variables from the recorded flight data. The number of random variables N is PC =(m+n)! / (m!n!); (2) N PC The sampling points are used as the input conditions for CFD simulation calculations, and the deterministic response results of each sampling point are calculated. i=0,1,…,n-1.

3. The method for obtaining the uncertainty of propeller slipstream effect based on arbitrary polynomial chaos expansion according to claim 1 is characterized in that: The specific steps of step 4 are as follows: The system response is expanded using a chaotic polynomial: where c0, c1, ..., c n-1 is the unknown coefficient, which can be obtained by the least squares method: The polynomial basis P (k) (x) is obtained from step 2, the system response of each sampling point From step 3, we can get the unknown coefficient c of the chaotic polynomial k Determine, thereby determining the chaotic polynomial f(x) of the entire system response.

4. The method for obtaining the uncertainty of propeller slipstream effect based on arbitrary polynomial chaos expansion according to claim 1, characterized in that: The specific steps of step 5 are as follows: (1) Compare the n-order and n+1-order results to determine whether the n-order result is close to the n+1-order result. If so, the accuracy of the n-order result is satisfactory. If the two results differ greatly, the order needs to be increased, and then the n+1-order and n+2-order results are compared. This process is repeated until the appropriate polynomial order is found. (2) After the order is determined, calculate N PC The correlation index between the configuration points and the chaotic polynomial: in is the residual sum of squares, is the total deviation square sum, judging the correlation coefficient R 2 Is it close to 1? If so, it means that the chaotic polynomial meets the requirements for the goodness of fit of the system. Otherwise, the number of sampling points N needs to be increased.

5. The method for obtaining the uncertainty of propeller slipstream effect based on arbitrary polynomial chaos expansion according to claim 1, characterized in that: The specific steps of step 6 are as follows: Substitute the N sampling points of the uncertainty variables in the propeller aircraft flight data into the chaotic polynomial of the system response Get the system response f(x) at each sampling point i ), i = 1, 2, ..., N, and statistically analyze the N system responses to obtain the probability distribution of the system responses caused by the uncertainty variables.

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