Non-stationary turbulent wind field large eddy simulation inlet synthesis method and device

Abstract

The application discloses a non-stationary turbulent wind field large eddy simulation entrance synthesis method and device, comprising the following steps: a continuous derivable sine wave is used to simulate a non-stationary time-varying average wind speed; a PRFG 3 The random flow method generates three-dimensional anisotropic turbulence with specific vertical wind shear parameters, turbulence intensity and turbulence integral scale; the sine wave and the random flow are superimposed to generate a non-stationary turbulent wind field with accurately adjustable non-stationary parameters. By adopting the technical scheme of the application, the sudden rise and fall of the time-varying average wind speed and the vertical wind shear characteristics can be accurately simulated, and the time-varying turbulence intensity and integral scale of the fluctuation component can also be accurately simulated, so that the application can be used for large eddy simulation research on strong typhoons, thunderstorm winds and downbursts and the like.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of computational fluid dynamics, and particularly relates to a non-stationary turbulent wind field large eddy simulation inlet synthesis method and device. BACKGROUND

[0002] With global warming, strong typhoons, thunderstorm winds and downbursts occur frequently, which have strong short-time sudden rise and sudden drop in the horizontal direction and show strong non-stationary characteristics, and show significant wind shear effects in the vertical direction. The load effect of the specific strong wind is significantly higher than that of the traditional stationary wind field, and the related wind-resistant design specification is not yet perfect, which seriously threatens the wind-resistant safety of large-span bridges, large offshore wind turbines, flexible photovoltaic supports and other new energy structures. At present, the research on non-stationary strong wind mainly relies on active wind tunnel tests, which need expensive impact jets or active turbulent flow generating devices, and it is urgent to develop a CFD simulation method for non-stationary shear wind field.

[0003] Large eddy simulation is widely used in computational fluid dynamics due to its advantages of considering calculation accuracy and cost. However, the current large eddy simulation research on non-stationary wind load still continues the impact jet model, which needs more than one million auxiliary grids to generate a non-stationary wind field, and it is difficult to accurately control the acceleration factor, vertical wind shear factor, time-varying turbulence intensity and turbulence integral scale and other non-stationary strong wind parameters, which limits the in-depth study of the non-stationary strong wind load effect. The large eddy simulation inlet turbulent flow synthesis method can significantly reduce the calculation cost and better control the turbulent characteristics, but the research on the large eddy simulation inlet synthesis turbulent flow for non-stationary wind field is less, and the traditional random flow (Random Flow Generation) synthesis method can only consider the stationary wind field inlet. Therefore, it is urgent to develop a high-efficiency and accurate large eddy simulation inlet method for non-stationary vertical shear wind field. SUMMARY

[0004] The technical problem to be solved by the present application is to provide a non-stationary turbulent wind field large eddy simulation inlet synthesis method and device based on sine wave superposition random flow.

[0005] To achieve the above-mentioned purpose, the technical scheme adopted by the present application is as follows:

[0006] A non-stationary turbulent wind field large eddy simulation inlet synthesis method, comprising:

[0007] Step S1, using a continuously derivable sine wave to simulate a time-varying non-stationary average wind speed;

[0008] Step S2, using PRFG 3 Random flow method to generate three-dimensional anisotropic turbulence with specific vertical wind shear parameters, turbulence intensity and turbulence integral scale;

[0009] Step S3, superimposing both the sine wave and the random stream to generate a non-stationary parameter-adjustable non-stationary turbulent wind field.

[0010] As preferred, in step S1, a non-stationary sinusoidal inflow is superimposed on the basis of a fixed uniform inflow, which is continuously derivable to meet the condition that no sudden wind speed fluctuation occurs, and the calculation formula of the wind field is:

[0011]

[0012] wherein U(t) is a three-dimensional time-varying average wind speed vector matrix, n is the number of superimposed sine waves, a i , b i and c i are parameters of each superimposed sine wave, and t is time.

[0013] As preferred, in step S2, the anisotropic turbulence is generated based on the PRFG 3 method, the vertical wind shear turbulent wind field with a specific wind profile and turbulence intensity profile is realized by introducing a weighting function and a phase difference, the linear convection based on the time-averaged velocity is used to approximate the momentum balance, and the velocity field satisfies the following equation:

[0014]

[0015] wherein u=[u,v,w] T is a three-dimensional fluctuating wind speed vector matrix, x=[x,y,z] is a three-dimensional coordinate vector, N is the number of sampling wave vectors, M is the number of turbulence source vectors, p n,m and q n,m are random amplitude vector matrices, k n,m is a random wave number matrix, ω n,m is a random fluctuating wind circular frequency, ψ n,m is a phase offset, w m (x) is a weighting function of each turbulence source, (·) T is a transpose operator.

[0016] As preferred, it further comprises: a variable-based inflow correction method for correcting the velocity components located on the inlet plane to alleviate the false pressure fluctuation caused.

[0017] The present application also provides a non-stationary turbulent wind field large eddy simulation inlet synthesis device, comprising:

[0018] A first calculation module is configured to simulate a time-varying non-stationary average wind speed by using a continuously derivable sine wave.

[0019] A second calculation module is configured to generate an anisotropic turbulence by using a PRFG 3The random stream method generates three-dimensional anisotropic turbulence with specific vertical wind shear parameters, turbulence intensity and turbulence integral scale;

[0020] The third calculation module is used for superimposing both the sine wave and the random stream to generate a non-stationary turbulence wind field with precisely adjustable non-stationary parameters.

[0021] As a preference, the first calculation module is used for superimposing a non-stationary sine inflow on the basis of a fixed uniform inflow, which is continuously derivable to meet the requirement that no sudden wind speed fluctuation occurs, and the calculation formula of the wind field is:

[0022]

[0023] wherein U(t) is a three-dimensional time-varying average wind speed vector matrix, n is the number of superimposed sine waves, a i , b i and c i are parameters of each superimposed sine wave, and t is time.

[0024] As a preference, the second calculation module is used for generating anisotropic turbulence based on a PRFG 3 method, vertical wind shear turbulence wind field with specific wind profile and turbulence intensity profile is realized by introducing a weighting function and a phase difference, linear convection based on time-averaged velocity is used to approximate momentum balance, and the velocity field satisfies the following equation:

[0025]

[0026] wherein u = [u, v, w] T is a three-dimensional fluctuating wind speed vector matrix, x = [x, y, z] is a three-dimensional coordinate vector, N is the number of sampling wave vectors, M is the number of turbulence source vectors, p n,m and q n,m are random amplitude vector matrices, k n,m is a random wave number matrix, ω n,m is a random fluctuating wind circular frequency, ψ n,m is a phase offset, w m (x) is a weighting function of each turbulence source, (·) T is a transpose operator.

[0027] As a preference, the fourth calculation module is used for correcting the velocity components located on the inlet plane based on a variable inflow correction method to relieve the false pressure fluctuation caused.

[0028] The present application firstly simulates non-stationary average wind speed and vertical wind shear through a sine wave, adopts a PRFG 3Realize anisotropic turbulence, then superimpose both to realize non-stationary turbulent wind field, and finally reduce the false fluctuation of static pressure wave at the entrance by VBIC method. The application can not only accurately simulate the sudden rise and sudden drop in the horizontal direction and the vertical wind shear characteristics, but also accurately adjust the time-varying turbulence and integral scale. The application can be used for large eddy simulation of non-stationary strong typhoon, thunderstorm wind and downburst, and compared with the large eddy simulation of impinging jet, the proposed method greatly reduces the calculation cost and improves the simulation accuracy, and has high application value in the fields of large-span bridge wind engineering, photovoltaic wind resistance and offshore wind power. BRIEF DESCRIPTION OF DRAWINGS

[0029] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments or the prior art description will be briefly introduced below. Obviously, the drawings in the following description are only a part of the embodiments of the present application, and all other drawings obtained by those skilled in the art without creative labor based on the provided drawings are within the protection scope of the present application.

[0030] Figure 1 It is a flowchart of the non-stationary turbulent wind field large eddy simulation inlet synthesis method of the present application.

[0031] Figure 2 It is a large eddy simulation calculation domain and grid division schematic diagram (unit: m), wherein (a) is a calculation domain schematic diagram, and (b) is an adopted grid schematic diagram.

[0032] Figure 3 It is a non-stationary wind field large eddy simulation process and result diagram generated by embodiment 1 of the present application.

[0033] Figure 4 It is a cable-stayed bridge structure measurement point arrangement schematic diagram (unit: cm) of the present application.

[0034] Figure 5 It is a large eddy simulation and measurement curve comparison diagram of the bridge measured non-stationary strong wind in embodiment 3 of the present application and the vertical wind shear non-stationary strong wind in embodiment 4, wherein (a) is a wind speed time history curve, and (b) is a downburst wind profile measured at the United States Anderson Air Force Base (AAFB). DETAILED DESCRIPTION

[0035] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor are within the protection scope of the present application.

[0036] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0037] Example 1:

[0038] This invention provides a method for synthesizing the inlet of a large eddy simulation of a non-stationary turbulent wind field. It employs a continuously differentiable sine wave to simulate the time-varying non-stationary average wind speed and a random flow generator of a specified wavelength. 3 (Prescribed-wavelengthRandom Flow Generator 3 PRFG 3 The random flow method generates three-dimensional anisotropic turbulence with specific vertical wind shear parameters, turbulence intensity, and turbulence integral scale, while the VBIC method reduces spurious pressure fluctuations common when using synthetic inlet flows. The superposition of sinusoidal waves and random flow generates a non-stationary turbulent wind field with precisely adjustable non-stationary parameters. The wind field in this embodiment of the invention, in situations such as... Figure 2 Generated in the computational domain grid shown in (a) and (b), as Figure 1 As shown, the inlet synthesis of large eddy simulation for non-stationary turbulent wind fields includes the following steps:

[0039] Step 1: Analysis of measured non-stationary strong wind characteristic parameters. Based on EMD decomposition, time-varying average wind speed, turbulence intensity, and turbulence integral scale parameters are obtained, specifically:

[0040]

[0041] Where: I(n) is the input signal, IMF m (n) is M th The intrinsic modulus function, Res M (n) represents the residual.

[0042] Step 2: Sine wave fitting modeling of non-stationary time-varying average wind speed: A non-stationary sinusoidal inflow is superimposed on a fixed uniform inflow. This inflow needs to be continuously differentiable to avoid sudden wind speed fluctuations. The formula for calculating this wind field is:

[0043]

[0044] Where: U(t) is the three-dimensional time-varying average wind speed vector matrix, n is the number of terms in the superimposed sine wave, and a i b i and c i Let t be the parameter of each superimposed sine wave, and t be the time.

[0045] Step three, applying vertical wind shear: Based on the measured non-stationary strong wind profile data, the wind profile expression of wind speed changing with height is constructed and compiled into the program, and the calculation formula of the profile is:

[0046]

[0047] wherein: U v is the wind speed changing with height, y m is the reference height, U m is the wind speed at the reference height, is the relationship formula between any height and the reference height.

[0048] Step four, generating anisotropic turbulence based on PRFG 3 method, the vertical wind shear turbulence wind field with specific wind profile and turbulence intensity profile is realized by introducing the weighting function and phase difference, the linear convection based on time-averaged velocity is used to approximate the momentum balance, which ensures that the synthesized turbulence field is as close as possible to the Navier-Stokes equation near the time-averaged condition, and the specific velocity field needs to satisfy the following equation:

[0049]

[0050] wherein: u=[u,v,w] T is the three-dimensional fluctuating wind speed vector matrix, x=[x,y,z] is the three-dimensional coordinate vector, N is the sampling wave vector number, M is the turbulence source vector number, p n,m and q n,m are random amplitude vector matrices, k n,m is a random wave number matrix, ω n,m is a random fluctuating wind circular frequency, ψ n,m is a phase shift, w m (x) is the weighting function of each turbulence source, (·) T is the transpose operator.

[0051] Step five, superimposing non-stationary time-varying average wind speed and random turbulence, for non-stationary incoming flow, the fluctuating component of the velocity field under the uniform flow needs to be superimposed in the x direction, so the velocity field under the non-stationary incoming flow is calculated as:

[0052]

[0053] wherein: u n-s =[u n-s ,v s ,w s ] T is the three-dimensional synthesized velocity field under the non-stationary incoming flow.

[0054] Step six, non-stationary wind field large eddy simulation inlet preparation, through the open source computing software OpenFOAM, the above non-stationary turbulence theory formula C++ programming, that is, the precise adjustment of non-stationary acceleration time, turbulence and turbulence integral scale and other strong wind characteristics can be realized.

[0055] Step seven, the entrance fluctuation false static pressure fluctuation correction based on the variable inflow correction (VBIC) method, since the synthetic turbulence field does not match the Navier-Stokes equation and the boundary condition adjacent to the inlet region, false pressure fluctuation is caused, based on the VBIC method, the velocity component located on the inlet plane is corrected to alleviate the false pressure fluctuation caused, the calculation formula is:

[0056] u = u s + u c (6)

[0057] Wherein: u is the corrected velocity field actually applied to the inlet region, u c = [0, v c , w c ] T Is the velocity correction term calculated to minimize the norm.

[0058] Example 2

[0059] The embodiment of the application provides a non-stationary turbulent wind field large eddy simulation inlet synthesis method based on sine wave superposition random flow, and the specific steps are as follows:

[0060] 1) generate a time-varying average wind of a sine flow

[0061] For the applied sine inflow, the formula of the time-varying average wind speed is as follows:

[0062]

[0063] In the formula, U(t) is the time-varying average wind speed along the x direction, U s is the initial wind speed, U e is the cut-off wind speed, t s is the acceleration start time, t e is the acceleration end time. In this example, t s = 2s, t e = 4s, U s = 3m / s, U e = 6m / s.

[0064] 2) generate fluctuating turbulence

[0065] For the applied pulsating turbulent flow, the wind speed expression is shown in equation (4), where the average wind speed adopted is U = 4.5 m / s.

[0066] 3) Sinusoidal flow superimposed with pulsating turbulent flow

[0067] The superimposition process and results are shown in Figure 3 .

[0068] Example 3:

[0069] Based on the non-stationary strong wind data measured by the cable-stayed bridge health monitoring system in Example 2, the large eddy inlet simulation is realized, and the specific steps are as follows:

[0070] 1) Cable-stayed bridge health monitoring system

[0071] Taking a certain super large cable-stayed bridge as the research object, as shown in Figure 4 , the bridge has a total length of 1339 m, a main span of 618 m, and is a double-tower double-cable-plane hybrid girder cable-stayed bridge with a semi-floating system structure of tower-girder separation. The tower is a diamond-shaped variable cross-section tower with a tower height of 216 m. The main girder of the main span adopts a steel box girder, and the main girder of the side span adopts a concrete girder. The main bridge deck is 31 m wide.

[0072] For wind speed monitoring point arrangement and real-time signal acquisition, three-dimensional anemometers are installed on the steel box girder section and the corresponding measuring points of the main tower at the positions shown in Figure 4 , which are used to measure the real-time changes of wind speed at different cross-section positions of the main girder of the cable-stayed bridge and at different heights of the main tower.

[0073] 2) Analysis of characteristics of measured non-stationary strong wind samples

[0074] By analyzing the wind field monitoring data, non-stationary strong wind samples are selected, and wind parameters such as time-varying average wind speed, turbulence intensity, turbulence integral scale, wind yaw angle, wind attack angle, and power spectrum are analyzed.

[0075] 3) Time-varying average wind of measured non-stationary strong wind

[0076] The following formula is used to represent the time-varying average wind of the measured non-stationary strong wind:

[0077]

[0078] In this example, a1 = 16.34, a2 = 2.239, a3 = 3.439, b1 = 0.0329, b2 = 0.185, b3 = 0.102, c1 = 0.178, c2 = -3.577, and c3 = -0.788.

[0079] 4) Large eddy simulation of measured non-stationary turbulent flow

[0080] The large eddy simulation of the measured non-stationary turbulent flow is compared with the measurement as shown in Figure 5As shown in (a) and (b).

[0081] Example 4:

[0082] Based on Examples 2 and 3, the simulation of non-stationary turbulent wind fields in downbursts with vertical wind shear is achieved. The specific steps are as follows:

[0083] 1) Selection of downburst profile

[0084] Using a typical downburst wind profile from Andersen Air Force Base (AAFB), the relationship between any altitude and a reference altitude is expressed by the following formula:

[0085]

[0086] In the formula: γ is an empirical constant. In this example, we set γ = 0.261, y m =10m.

[0087] 2) Generation of downburst profiles

[0088] Through a non-stationary wavelength random flow generator 3 (PRFG 3 The wind parameters analyzed above were compiled and the selected non-stationary turbulent wind field was simulated. The simulated wind parameters were calculated and compared with those in the actual measured analysis to verify the accuracy of the simulation. A comparison of the simulated and measured curves of the actual non-stationary strong winds is shown below. Figure 5 As shown; a comparison between simulated and measured large eddy currents of storm shear at Andersen Air Force Base (AAFB), USA. Figure 5 As shown.

[0089] Example 5:

[0090] This invention also provides an inlet synthesis device for simulating large eddies in non-stationary turbulent wind fields, comprising:

[0091] The first calculation module is used to simulate time-varying non-stationary average wind speed using a continuously differentiable sine wave;

[0092] The second calculation module is used to employ PRFG. 3 The random flow method generates three-dimensional anisotropic turbulence with specific vertical wind shear parameters, turbulence intensity, and turbulence integral scale;

[0093] The third calculation module is used to superimpose the sine wave and the random flow to generate a non-stationary turbulent wind field with precisely adjustable non-stationary parameters.

[0094] In one embodiment of the present invention, the first calculation module is used to superimpose a non-stationary sinusoidal incoming flow on a fixed uniform incoming flow. The incoming flow is continuously differentiable to ensure that no sudden wind speed fluctuations occur. The calculation formula for this wind field is:

[0095]

[0096] where U(t) is a three-dimensional time-varying mean wind speed vector matrix, n is the number of superimposed sinusoidal waves, a i , b i and c i are parameters of each superimposed sinusoidal wave, and t is time.

[0097] As an embodiment of the present application, the second calculation module is configured to generate anisotropic turbulence based on a PRFG 3 method, a vertical wind shear turbulence wind field with a specific wind profile and turbulence intensity profile is realized by introducing a weighting function and a phase difference, a linear convection based on a time-averaged velocity is used to approximate a momentum balance, and the velocity field satisfies the following equation:

[0098]

[0099] where u = [u, v, w] T is a three-dimensional fluctuating wind speed vector matrix, x = [x, y, z] is a three-dimensional coordinate vector, N is the number of sampling wave vectors, M is the number of turbulence source vectors, p n,m and q n,m are random amplitude vector matrices, k n,m is a random wave number matrix, ω n,m is a random fluctuating wind circular frequency, ψ n,m is a phase offset, w m (x) is a weighting function of each turbulence source, and (·) T is a transpose operator.

[0100] As an embodiment of the present application, the fourth calculation module is further configured to correct the velocity components located on the inlet plane based on a variable inflow correction method to alleviate the false pressure fluctuations caused.

[0101] The above-described embodiments are merely descriptions of the preferred modes of the present application, and are not intended to limit the scope of the present application. Various modifications and improvements to the technical solutions of the present application made by those of ordinary skill in the art without departing from the design spirit of the present application shall fall within the scope of protection of the present application as defined by the claims.

Claims

1. A method for synthesizing the inlet of a large eddy simulation of a non-stationary turbulent wind field, characterized in that, include: Step 1: Measure the non-stationary strong wind characteristic parameters and obtain the time-varying average wind speed, turbulence intensity, and turbulence integral scale parameters based on EMD decomposition; Step 2: Sine wave fitting modeling of non-stationary time-varying average wind speed. A non-stationary sinusoidal inflow is superimposed on a fixed uniform inflow. The inflow is continuous and differentiable to ensure that there are no sudden wind speed fluctuations. Step 3: Apply vertical wind shear and construct a wind profile showing the variation of wind speed with height based on measured non-stationary strong wind profile data; Step 4: Based on PRFG 3 The method generates anisotropic turbulence and realizes vertical wind shear turbulent wind field with wind profile and turbulence intensity profile by introducing weighting function and phase difference. The momentum balance is approximated by linear convection based on time-averaged velocity. Step 5: Superimpose non-stationary time-varying average wind speed with random turbulence. For non-stationary incoming flow, superimpose the fluctuating component of the uniform flow velocity field in the x-direction to generate a non-stationary turbulent wind field with precisely adjustable non-stationary parameters.

2. The inlet synthesis method for large eddy simulation of non-stationary turbulent wind fields as described in claim 1, characterized in that, In step two, a non-stationary sinusoidal inflow is superimposed on a fixed, uniform inflow. This inflow is continuously differentiable to prevent sudden wind speed fluctuations. The formula for calculating this wind field is: Where U(t) is the three-dimensional time-varying average wind speed vector matrix, n is the number of terms in the superimposed sine wave, and a i b i and c i Let t be the parameter of each superimposed sine wave, and t be the time.

3. The inlet synthesis method for large eddy simulation of non-stationary turbulent wind fields as described in claim 2, characterized in that, In step four, based on PRFG 3 The method generates anisotropic turbulence and realizes a vertical wind shear turbulent wind field with wind profile and turbulence intensity profile by introducing a weighting function and a phase difference. The momentum balance is approximated by linear convection based on time-averaged velocity, and the velocity field satisfies the following equation: Among them, u s =[u,v,w] T Let x = [x, y, z] be the three-dimensional fluctuating wind speed vector matrix, N be the number of sampled wave vectors, M be the number of turbulence source vectors, and p be the number of turbulence source vectors. n,m and q n,m Let k be a random magnitude vector matrix. n,m Let ω be a random wavenumber matrix. n,m Let ψ be the circumferential frequency of the random pulsating wind. n,m For phase shift, w m (x) is the weighting function for each turbulence source, (·) T This is the transpose operator.

4. The inlet synthesis method for large eddy simulation of non-stationary turbulent wind fields as described in claim 3, characterized in that, Also includes: Variable-based inflow correction methods correct the velocity components located on the inlet plane to mitigate spurious pressure fluctuations.

5. A device for synthesizing the inlet of a non-stationary turbulent wind field using large eddy simulation as described in claim 1, characterized in that, include: The first calculation module is used to simulate time-varying non-stationary average wind speed using a continuously differentiable sine wave; The second calculation module is used to employ PRFG. 3 The random flow method generates three-dimensional anisotropic turbulence with vertical wind shear parameters, turbulence intensity, and turbulence integral scale; The third calculation module is used to superimpose the sine wave and the random flow to generate a non-stationary turbulent wind field with precisely adjustable non-stationary parameters.

6. The inlet synthesis device for simulating large eddies in non-stationary turbulent wind fields as described in claim 5, characterized in that, The first calculation module is used to superimpose a non-stationary sinusoidal inflow onto a fixed, uniform inflow. This inflow is continuously differentiable to prevent sudden wind speed fluctuations. The calculation formula for this wind field is: Where U(t) is the three-dimensional time-varying average wind speed vector matrix, n is the number of terms in the superimposed sine wave, and a i b i and c i Let t be the parameter of each superimposed sine wave, and t be the time.

7. The inlet synthesis device for large eddy simulation of non-stationary turbulent wind fields as described in claim 6, characterized in that, The second calculation module is used based on PRFG. 3 The method generates anisotropic turbulence and realizes a vertical wind shear turbulent wind field with wind profile and turbulence intensity profile by introducing a weighting function and a phase difference. The momentum balance is approximated by linear convection based on time-averaged velocity, and the velocity field satisfies the following equation: Among them, u s =[u,v,w] T Let x = [x, y, z] be the three-dimensional fluctuating wind speed vector matrix, N be the number of sampled wave vectors, M be the number of turbulence source vectors, and p be the number of turbulence source vectors. n,m and q n,m Let k be a random magnitude vector matrix. n,m Let ω be a random wavenumber matrix. n,m Let ψ be the circumferential frequency of the random pulsating wind. n,m For phase shift, w m (x) is the weighting function for each turbulence source, (·) T This is the transpose operator.

8. The inlet synthesis device for large eddy simulation of non-stationary turbulent wind fields as described in claim 7, characterized in that, Also includes: The fourth calculation module is used to correct the velocity component located on the inlet plane using a variable-based inflow correction method to mitigate spurious pressure fluctuations.

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