Two-dimensional vortex wave numerical synthesis method based on non-divergence momentum source

Abstract

The invention discloses a two-dimensional vortex wave numerical synthesis method based on a non-divergence momentum source, and relates to the technical field of fluid mechanics, and the method comprises the steps: constructing a two-dimensional linear momentum equation in a source item assumption state according to the boundary information of a calculation region; fourier transform in a time domain is carried out on the two-dimensional linear momentum equation, and a first-order ordinary differential equation in a frequency domain is obtained; solving the first-order ordinary differential equation to obtain a relational expression between a complex variable of the induced velocity field and a source item; based on the constraint of the induced velocity field corresponding to the upstream and downstream of the region boundary, obtaining the compatibility condition of the downstream induced velocity field according to the relational expression; obtaining a source item expression by deducing the field divergence of the induced velocity under the constraint of a compatibility condition and a non-divergence condition; and according to the source item expression and the target two-dimensional vortex wave disturbance, synthesizing and obtaining a source item value. According to the method, an ideal vortex wave synthesis result is obtained under the condition that no scattering is guaranteed, and the reliability and stability of fluid mechanics research are improved.

Description

Technical Field

[0001] The invention relates to the technical field of fluid mechanics, and in particular to a two-dimensional vortex wave numerical synthesis method based on a divergence-free momentum source. Background Art

[0002] Turbulence is an important research object in fluid mechanics and is of great significance to major engineering applications such as aircraft, helicopters and submarines. In practical problems, due to the strong randomness of turbulence, direct application of numerical simulation to study complex fluid mechanics problems usually faces difficulties such as poor accuracy and low efficiency.

[0003] Vortex waves are the basic components of turbulence. Introducing specific turbulent structures into the numerical simulation calculation area by artificially synthesizing vortex waves and making them interact with airfoils, blades or cascade structures has become one of the most popular research areas in turbulence research. Among them, there are usually two methods for synthesizing vortex waves, one is introduced by the boundary conditions at the inlet of the calculation domain, and the other is synthesized in the calculation domain by adding a momentum source (source term).

[0004] On the one hand, the existing method of synthesizing vortex waves through boundary conditions cannot ensure that the velocity field satisfies the divergence-free condition, and numerical pseudo-noise will be generated at the boundary. It is not easy to be compatible with the existing boundary conditions and faces the disadvantage of high computational cost. On the other hand, the existing method of synthesizing vortex waves in the computational domain by adding momentum sources cannot meet the velocity divergence-free condition near the added source area, and will also produce numerical pseudo-noise problems, which will affect the reliability and stability of numerical simulation. Summary of the invention

[0005] In order to provide a vortex wave synthesis method with higher accuracy and higher efficiency, the present invention proposes a two-dimensional vortex wave numerical synthesis method based on a divergence-free momentum source, comprising the steps of:

[0006] S1: Based on the boundary information of the calculation area, construct the two-dimensional linear momentum equation under the source term assumption state;

[0007] S2: Obtain the first-order ordinary differential equation in the frequency domain by performing Fourier transform on the two-dimensional linear momentum equation in the time domain;

[0008] S3: By solving the first-order ordinary differential equation, the relationship between the complex variable of the induced velocity field and the source term is obtained;

[0009] S4: Based on the constraints of the corresponding induced velocity fields upstream and downstream of the regional boundary, the compatibility conditions of the downstream induced velocity field are obtained according to the relationship;

[0010] S5: Under the constraints of compatibility conditions and divergence-free conditions, the source term expression is obtained by deriving the divergence of the induced velocity field;

[0011] S6: Synthesize and obtain the source term value based on the source term expression and the target two-dimensional eddy wave disturbance.

[0012] Furthermore, in step S1, the two-dimensional linear momentum equation is expressed as the following formula:

[0013]

[0014]

[0015] Where u is the horizontal flow velocity, v is the vertical flow velocity, t is the time parameter, x is the horizontal coordinate of the calculation area, and u is ∞ is the uniform velocity of the incoming flow, (S u , S v ) is the assumed source term, S u is the horizontal momentum source, S v is the vertical momentum source.

[0016] Furthermore, in step S2, the first-order ordinary differential equation is expressed as the following formula:

[0017]

[0018]

[0019] In the formula, u g is the horizontal pulsating velocity of the source term, v g is the vertical pulsation velocity of the source term, the superscript ~ represents the frequency domain component, i is the imaginary unit, i 2 =-1, k x is the spatial wave number of the fundamental mode along the horizontal direction, k x =ω / u ∞ , ω is the angular frequency of the simple harmonic oscillation of the source term.

[0020] Furthermore, in step S3, the relationship between the induced velocity field complex variable and the source term is obtained by the following formula:

[0021]

[0022]

[0023]

[0024] In the formula, R is the operator for taking the real part, e is the natural logarithm, ξ is the temporary integrand, and C u and C v is a constant, For S u The frequency domain components of For S v The frequency domain components of .

[0025] Furthermore, in the step S4, the constraints of the induced velocity fields corresponding to the upstream and downstream of the region boundary are specifically: the upstream induced velocity field is 0, and the downstream induced velocity field is the target velocity field.

[0026] Furthermore, in the step S4, the compatibility condition is expressed as the following formula:

[0027]

[0028]

[0029] In the formula, k y is the spatial wave number in the vertical direction of the fundamental mode, ε u is the amplitude of the horizontal velocity pulsation, ε v is the amplitude of velocity pulsation in the vertical direction, and y is the vertical coordinate of the calculation area.

[0030] Furthermore, in the step S5, obtaining the source term expression specifically includes the following steps:

[0031] S51: Obtain the divergence expression of the induced velocity field, and obtain the upstream establishment condition of the region under the constraint of the upstream induced velocity field;

[0032] S52: Under the condition that the source terms in the frequency domain and the time domain are divergent and equivalent, the frequency domain expression of the source term is obtained by introducing the compatibility condition;

[0033] S53: Obtain the source term expression by substituting the divergence condition.

[0034] Furthermore, in the step S51, the induced velocity field divergence expression is as follows:

[0035]

[0036]

[0037] Under the constraint that the upstream induced velocity field is 0, f(x) = 0, that is:

[0038]

[0039] Get the regional upstream establishment conditions

[0040] Furthermore, in the step S52, the frequency domain expression of the source term is obtained by the following formula:

[0041]

[0042]

[0043]

[0044]

[0045] Where g(x) and h(x) are distribution functions, A u and A v is an undetermined constant, where g(x) is a custom distribution function. h(x) and A are derived from g(x). u and A v The solution.

[0046] Furthermore, in the step S53, the source term expression is expressed as the following formula:

[0047]

[0048]

[0049] Where ∈ is the velocity pulsation amplitude, and G(x) is the integral transformation related to g(x).

[0050] Compared with the prior art, the present invention has at least the following beneficial effects:

[0051] The present invention discloses a two-dimensional vortex wave numerical synthesis method based on a divergence-free momentum source. By combining the introduction of boundary conditions at the entrance of the computational domain with the addition of a momentum source in the computational domain, the two-dimensional linearized momentum equation of the assumed source term is reversely deduced by the boundary conditions, and the expression of the source term is obtained in combination with the divergence-free condition. Thus, the source term obtained by the expression can obtain an ideal vortex wave synthesis result while ensuring divergence-free, thereby greatly improving the reliability and stability of fluid mechanics research. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is a step diagram of a two-dimensional eddy wave numerical synthesis method based on a divergence-free momentum source;

[0053] Figure 2 Schematic diagram of an example of synthetic turbulence and airfoil interference. DETAILED DESCRIPTION

[0054] The following are specific embodiments of the present invention and the accompanying drawings to further describe the technical solution of the present invention, but the present invention is not limited to these embodiments.

[0055] Embodiment 1

[0056] In order to describe the technical points of the present invention more vividly, Figure 2As shown in FIG. 1 , it is a schematic diagram of the implementation of the present invention (airfoil interference simulation), in which the calculation area is represented by a rectangular boundary (in actual situations, linear transformation can be used to adapt to different boundary shapes), the upstream of the boundary (Inlet, the left side of the calculation area) is the velocity inlet boundary, the downstream (Outlet, the right side of the calculation area) is the pressure outlet condition, and the rest (the upper and lower sides of the calculation area) are far-field characteristic boundary conditions. Without loss of generality, we assume that the uniform fluid flows from left to right with u ∞ The present invention moves at a horizontal speed (vertical speed is 0). The present invention calculates the area (width W and height H, the center coordinate position is (x s ,y s )) Add a suitable source region, that is, a two-dimensional momentum source (S u , S v ), so that the two-dimensional eddy wave disturbance (u ∞ +u g , v ∞ +v g ), where u g and v g are the horizontal and vertical pulsation velocities of the source term, respectively.

[0057] Considering that in many fluid mechanics problems, turbulent flow can be decomposed into uniform average flow and small-amplitude turbulent pulsation, and turbulent pulsation satisfies Taylor's freezing hypothesis and convects with the average flow without changing its shape. On this basis, Figure 1 As shown, the present invention proposes a two-dimensional vortex wave numerical synthesis method based on a divergence-free momentum source, comprising the steps of:

[0058] S1: Based on the boundary information of the calculation area, construct the two-dimensional linear momentum equation under the source term assumption state;

[0059] S2: Obtain the first-order ordinary differential equation in the frequency domain by performing Fourier transform on the two-dimensional linear momentum equation in the time domain;

[0060] S3: By solving the first-order ordinary differential equation, the relationship between the complex variable of the induced velocity field and the source term is obtained;

[0061] S4: Based on the constraints of the corresponding induced velocity fields upstream and downstream of the regional boundary, the compatibility conditions of the downstream induced velocity field are obtained according to the relationship;

[0062] S5: Under the constraints of compatibility conditions and divergence-free conditions, the source term expression is obtained by deriving the divergence of the induced velocity field;

[0063] S6: Synthesize and obtain the source term value based on the source term expression and the target two-dimensional eddy wave disturbance.

[0064] In order to make the analysis more convenient and adapt to disturbances in more complex situations (complex disturbances can be formed by the superposition of multiple small disturbances), here we use the two-dimensional linearized momentum equation based on small disturbances:

[0065]

[0066]

[0067] Where u is the horizontal flow velocity, v is the vertical flow velocity, t is the time parameter, x is the horizontal coordinate of the calculation area, and u is ∞ is the uniform velocity of the incoming flow, (S u , S v ) is the assumed source term (that is, the momentum source term that needs to be introduced), S u is the horizontal momentum source, S v is the vertical momentum source.

[0068] To solve this equation, we need to summarize u g and v g First, the two-dimensional linearized momentum equation is Fourier transformed to convert it into a first-order ordinary differential equation in the frequency domain. The first-order ordinary differential equation can be solved analytically, and then the inverse transformation can be performed to obtain the velocity field in the time domain (that is, the target velocity field).

[0069] Among them, the Fourier transform of the two-dimensional linear momentum equation in the time domain can be obtained:

[0070]

[0071]

[0072] In the formula, u g is the horizontal pulsating velocity of the source term, v g is the vertical pulsation velocity of the source term, the superscript ~ represents the frequency domain component, i is the imaginary unit, i 2 =-1, k x is the spatial wave number of the fundamental mode along the horizontal direction, k x =ω / u ∞ , ω is the angular frequency of the simple harmonic oscillation of the source term (which can be set according to the actual vortex wave research requirements).

[0073] Because more complex incoming flows can be obtained by linearly superimposing sinusoidal waves of multiple frequencies through the Fourier decomposition method, this example only considers the injection of a sinusoidal wave of one frequency. The relationship between the frequency domain component and the time domain component can be expressed as the following formula:

[0074]

[0075] Where R is the real part operator, and e is the natural logarithm.

[0076] After that, we first solve the previous ordinary differential equation and obtain the relationship between the induced velocity field complex variable (frequency domain component) and the source term as follows:

[0077]

[0078]

[0079] In the formula, ξ is the temporary variable to be integrated, C u and C v is a constant, For S u The frequency domain components of For S v The frequency domain components of .

[0080] At this point, we consider that the upstream of the calculation area needs to meet the requirement that the induced velocity field is 0, so we can confirm that the constant C u =C v = 0. In addition, the velocity field induced in the downstream of the calculation area should be the required target velocity field:

[0081]

[0082]

[0083] Therefore, under the constraint of the target velocity field, the induced velocity field in the downstream must satisfy the compatibility condition:

[0084]

[0085]

[0086] In the formula, k y is the spatial wave number in the vertical direction of the fundamental mode, ε u is the amplitude of the horizontal velocity pulsation, ε v is the amplitude of velocity pulsation in the vertical direction, and y is the vertical coordinate of the calculation area.

[0087] The divergence-free condition is ensured by the following derivation: First, we derive the induced velocity field (u g , v g ) to find the divergence:

[0088]

[0089] in,

[0090]

[0091] Since the source term upstream of the calculation area is required to be 0, we can get f(x) = 0. At this time, we only need to satisfy the following formula:

[0092]

[0093] That is, the establishment conditions It can be established in the calculation area. At the same time, due to Therefore, the source term in the frequency domain is equivalent to the source term in the time domain. Without loss of generality, it is assumed that the source term has the following form:

[0094]

[0095]

[0096] Where g(x) and h(x) are distribution functions, A u and A v is an undetermined constant, where g(x) is a custom distribution function. h(x) and A are derived from g(x). u and A v The solution.

[0097] Substituting the compatibility condition, we get:

[0098]

[0099]

[0100] Substituting the divergence-free condition, we can obtain the relationship between g(x) and h(x):

[0101]

[0102] So there is,

[0103]

[0104]

[0105] According to the above derivation, the specific expression of the two-dimensional eddy wave momentum source term is obtained as follows:

[0106]

[0107]

[0108] Where ∈ is the velocity disturbance amplitude, G(x) is the integral variable related to g(x), and ∈ is derived from the following formula:

[0109]

[0110]

[0111] In no scattered: Under the premise of x ε u +k y ε v =0, so the velocity disturbance amplitude is obtained by solving the following formula:

[0112]

[0113] in, Equivalent to ∈.

[0114] Furthermore, the distribution function g(x) and its derivative g′(x) along the x direction are defined as follows:

[0115]

[0116]

[0117] Furthermore, G(x) is defined as the integral transform with respect to g(x):

[0118]

[0119] In summary, the two-dimensional vortex wave numerical synthesis method based on the divergence-free momentum source described in the present invention combines the introduction of the boundary conditions at the entrance of the calculation domain with the addition of the momentum source in the calculation domain, reversely derives the two-dimensional linearized momentum equation of the assumed source term by the boundary conditions, and obtains the expression of the source term in combination with the divergence-free condition, so that the source term obtained by the expression can ensure divergence, and at the same time, under the condition of selecting a suitable momentum source addition area and setting its length and width parameters, the momentum source S required to be added at the current moment is calculated according to the given target vortex wave. u and S v The amplitude is obtained and added to the right side of the discrete momentum equation, and time-advancing iteration is performed to finally synthesize the target vortex wave downstream of the momentum source area, which greatly improves the reliability and stability of fluid mechanics research and avoids the generation of numerical pseudo-noise.

[0120] It should be noted that all directional indications in the embodiments of the present invention (such as up, down, left, right, front, back, etc.) are only used to explain the relative position relationship, movement status, etc. between the components under a certain specific posture (as shown in the accompanying drawings). If the specific posture changes, the directional indication will also change accordingly.

[0121] In addition, in the present invention, descriptions such as "first", "second", "one", etc. are only used for descriptive purposes and cannot be understood as indicating or implying their relative importance or implicitly indicating the number of the indicated technical features. Therefore, the features defined as "first" or "second" may explicitly or implicitly include at least one of the features. In the description of the present invention, the meaning of "plurality" is at least two, such as two, three, etc., unless otherwise clearly and specifically defined.

[0122] In the present invention, unless otherwise clearly specified and limited, the terms "connection", "fixation", etc. should be understood in a broad sense. For example, "fixation" can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium, it can be the internal connection of two elements or the interaction relationship between two elements, unless otherwise clearly defined. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0123] In addition, the technical solutions between the various embodiments of the present invention can be combined with each other, but it must be based on the fact that ordinary technicians in the field can implement it. When the combination of technical solutions is contradictory or cannot be implemented, it should be deemed that such combination of technical solutions does not exist and is not within the scope of protection required by the present invention.

Claims

1. A two-dimensional eddy wave numerical synthesis method based on a divergence-free momentum source, characterized in that: Includes steps: S1: Based on the boundary information of the calculation area, construct the two-dimensional linear momentum equation under the source term assumption state; S2: Obtain the first-order ordinary differential equation in the frequency domain by performing Fourier transform on the two-dimensional linear momentum equation in the time domain; S3: By solving the first-order ordinary differential equation, the relationship between the complex variable of the induced velocity field and the source term is obtained; S4: Based on the constraints of the corresponding induced velocity fields upstream and downstream of the regional boundary, the compatibility conditions of the downstream induced velocity field are obtained according to the relationship; S5: Under the constraints of compatibility conditions and divergence-free conditions, the source term expression is obtained by deriving the divergence of the induced velocity field; S6: Synthesize and obtain the source term value based on the source term expression and the target two-dimensional eddy wave disturbance.

2. A two-dimensional eddy wave numerical synthesis method based on a divergence-free momentum source as claimed in claim 1, characterized in that: In the step S1, the two-dimensional linear momentum equation is expressed as the following formula: Where u is the horizontal flow velocity, v is the vertical flow velocity, t is the time parameter, x is the horizontal coordinate of the calculation area, and u is ∞ is the uniform velocity of the incoming flow, (S u , S v ) is the assumed source term, S u is the horizontal momentum source, S v is the vertical momentum source.

3. A two-dimensional eddy wave numerical synthesis method based on a divergence-free momentum source as claimed in claim 2, characterized in that: In step S2, the first-order ordinary differential equation is expressed as the following formula: In the formula, u g is the horizontal pulsating velocity of the source term, v g is the vertical pulsation velocity of the source term, the superscript ~ represents the frequency domain component, i is the imaginary unit, i 2 =-1, k x is the spatial wave number of the fundamental mode along the horizontal direction, k x =ω / u ∞ , ω is the angular frequency of the simple harmonic oscillation of the source term.

4. A two-dimensional eddy wave numerical synthesis method based on a divergence-free momentum source as claimed in claim 3, characterized in that: In the step S3, the relationship between the induced velocity field complex variable and the source term is obtained by the following formula: In the formula, R is the operator for taking the real part, e is the natural logarithm, ξ is the temporary integrand, and C u and C v is a constant, For S u The frequency domain components of For S v The frequency domain components of .

5. A two-dimensional eddy wave numerical synthesis method based on a divergence-free momentum source as claimed in claim 4, characterized in that: In step S4, the constraints of the induced velocity fields corresponding to the upstream and downstream of the region boundary are specifically: the upstream induced velocity field is 0, and the downstream induced velocity field is the target velocity field.

6. A two-dimensional eddy wave numerical synthesis method based on a divergence-free momentum source as claimed in claim 5, characterized in that: In the step S4, the compatibility condition is expressed as the following formula: In the formula, k y is the spatial wave number in the vertical direction of the fundamental mode, ε u is the amplitude of the horizontal velocity pulsation, ε v is the amplitude of velocity pulsation in the vertical direction, and y is the vertical coordinate of the calculation area.

7. A two-dimensional eddy wave numerical synthesis method based on a divergence-free momentum source as claimed in claim 6, characterized in that: In the step S5, obtaining the source term expression specifically includes the following steps: S51: Obtain the divergence expression of the induced velocity field, and obtain the upstream establishment condition of the region under the constraint of the upstream induced velocity field; S52: Under the condition that the source terms in the frequency domain and time domain are divergent and equivalent, the frequency domain expression of the source term is obtained by introducing the compatibility condition; S53: Obtain the source term expression by substituting the divergence condition.

8. A two-dimensional eddy wave numerical synthesis method based on a divergence-free momentum source as claimed in claim 7, characterized in that: In the step S51, the induced velocity field divergence expression is as follows: Under the constraint that the upstream induced velocity field is 0, f(x) = 0, that is: Get the regional upstream establishment conditions 9. A two-dimensional eddy wave numerical synthesis method based on a divergence-free momentum source as claimed in claim 8, characterized in that: In the step S52, the frequency domain expression of the source term is obtained by the following formula: Where g(x) and h(x) are distribution functions, A u and A v is an undetermined constant, where g(x) is a custom distribution function. h(x) and A are derived from g(x). u and A v The solution.

10. A two-dimensional eddy wave numerical synthesis method based on a divergence-free momentum source as claimed in claim 9, characterized in that: In the step S53, the source term expression is expressed as the following formula: Where ∈ is the velocity disturbance amplitude, and G(x) is the integral transform related to g(x).

Images