JP5023778B2 - Large amplitude sloshing behavior prediction method - Google Patents

Description

  The present invention is used for predicting the behavior of a propellant tank of a spacecraft when a large-amplitude sloshing that causes the liquid inside the tank to move in a wide range due to the influence of a change in the direction of gravity occurs. The present invention relates to an amplitude sloshing behavior prediction method.

  As a technique for predicting the behavior when the liquid in the low gravity environment tank such as the propellant in the propellant tank of the spacecraft causes sloshing, the present inventor has so far proposed a tank 1 as shown in FIG. The linear vibration around the static equilibrium position is analyzed (for example, see Non-Patent Document 1 and Non-Patent Document 2), and the mechanical model (sloshing) is analyzed. Has proposed a method of creating a force and moment acting on the tank 1 by a mass spring system (see Non-Patent Document 3, for example). That is, specifically, the liquid level 3 at the time of static equilibrium of the propellant tank 1 in the spacecraft is not a plane perpendicular to the Z axis because it is intended for the case where surface tension is important in a low-gravity universe. The curved surface is axisymmetric with respect to the Z axis. The characteristics of the analysis proposed by the inventor in the past are as follows. As shown in FIG. 4, a conical contact with the tank wall surface at the contact line 4 between the liquid surface 3 and the tank wall surface during static equilibrium is considered. The spherical coordinates are set with the vertex O of the cone as the origin. This feature makes it possible to analytically determine the liquid motion characteristic function (mode function) for any axially symmetric tank shape with an outward convex shape, and to significantly reduce the calculation time cost. is there.

  As another method for predicting the sloshing behavior of a large amplitude propellant in a propellant tank of a spacecraft, a method belonging to a numerical analysis method using CFD has been proposed (for example, (See Patent Document 4).

Masahiko Utsumi (Low-gravity Propellant Slosh Analysis Using Spherical Coordinates), Journal of Fluids and Structures, (USA ), 1998, Vol. 12, p. 57-83 Masahiko Utsumi, “Low-gravity Sloshing in an Axisymmetrical Container Excited in the Axial Direction”, Journal of Applied Mechanics (Journal of Applied Mechanics) Mechanics), (USA), American Society of Mechanical Engineers (ASME), 2000, 67, p. 344-354 Utsumi M., “A Mechanical Model for Low-gravity Sloshing in an Axisymmetric Tank, Journal of Applied Mechanics, (USA) , American Society of Mechanical Engineers (ASME), 2004, 71, 724-730. Berry, LJ Demchak, JR Tegart, and MK Craig “An Analytical Tool for Simulating Large Amplitude Propellant Slosh”, AIAA Paper, (USA), American Aerospace Society (AIAA), 1981, No. 81-0500, p. 55-61

  However, analytical methods such as those described in Non-Patent Documents 1, 2, and 3 described above are used for behavior analysis when the liquid in the tank vibrates with a small amplitude or a finite amplitude near the static equilibrium position of the liquid surface. Is effective, but if a large-amplitude sloshing occurs where the direction of gravity changes and the liquid moves in the tank over a wide range, an application limit occurs. Although it is possible to predict the behavior of the liquid level in the initial slight time range, it is actually difficult to predict when the amplitude is increased.

  The method described in Non-Patent Document 4 requires a lot of calculation time and increases the cost. Further, since the CFD program is black boxed, it falls within a certain assumed range. There is also a possibility that a problem that behavior prediction can be performed only for sloshing may occur.

  Therefore, the present inventor has developed the analytical method proposed in Non-Patent Documents 1, 2, and 3 to predict the behavior of large-amplitude sloshing, such as when the liquid moves greatly from the static equilibrium position due to sudden gravity change. If the liquid level displacement from the initial position increases as a result of repeated efforts and research, the liquid level reference position is updated from the initial static equilibrium position to near the liquid level at that time and analyzed. For example, the present invention finds that the problem of large amplitude sloshing can be solved by repeating vibration analysis near the static equilibrium position proposed by the present inventor without using the CFD. Made.

  Therefore, the object of the present invention is to predict sloshing behavior with a large amplitude such that the liquid in the tank moves in the tank over a wide range as the gravity direction changes, such as a propellant tank of a spacecraft. Moreover, it is an object of the present invention to provide a large amplitude sloshing behavior prediction method capable of significantly reducing calculation time and cost.

In order to solve the above-described problems, the present invention uses a computer to perform a motion analysis near the reference position of the liquid level using the initial position at which the liquid in the tank is in a static equilibrium state as a reference position. When the displacement of the liquid level in the tank from the reference liquid level reaches 1/10 of the radial dimension of the tank , the liquid level is displaced at that time. And determining the time interval until the reference position of the initial liquid level is updated to the position of the new reference liquid level, and then the updated new level. A step of obtaining a principal axis of inertia of the reference liquid level, and taking an angular coordinate around the axis with the inertial main axis orthogonal to the updated reference liquid surface as an axis, the circumferential angle coordinate And divide the reference liquid level after the above update in each divided section. A tangent is drawn to the tank wall from the contact line of intersection between click wall, a step of determining the spherical coordinates of the liquid surface after the liquid surface displacement of the intersection of the該接line and the axis as a new starting point, the spherical coordinate is obtained Under the above conditions, for the time interval , the liquid surface motion analysis near the position of the new reference liquid surface is performed using the eigenmode of the liquid surrounded by the reference liquid surface and the tank wall surface, When the displacement from the reference liquid level reaches 1/10 of the radial dimension of the tank, the above steps are sequentially repeated, and the liquid level position at the end of the time interval is obtained, so that the position from the initial liquid level reference position is obtained. A large-amplitude sloshing behavior prediction method for predicting a large-amplitude sloshing behavior is provided.

According to the large-amplitude sloshing behavior prediction method of the present invention , the motion analysis near the reference position of the liquid level is performed by the computer using the initial position where the liquid in the tank is in a static equilibrium state as the reference position. When the displacement of the liquid level in the tank from the reference liquid level reaches 1/10 of the radial dimension of the tank , the liquid level displacement at that time is performed using the natural mode of the liquid surrounded by the tank wall surface. The subsequent liquid level is set as the position of the new reference liquid level, and a step for obtaining a time interval until the reference position of the initial liquid level is updated to the position of the new reference liquid level is then updated. A step of obtaining a principal axis of inertia of a new reference liquid surface, and taking a circumferential angle coordinate around the axis with the inertial principal axis orthogonal to the updated reference liquid surface as an axis, the circumferential angle Equally divide the coordinates, and after the update A tangent is drawn to the tank wall from the contact line of intersection between the tank wall of the quasi-liquid surface, a step of determining the spherical coordinates of the liquid surface after the liquid surface displacement of the intersection of the該接line and the axis as a new starting point, the ball Under the conditions for which the coordinates are obtained , the liquid level motion analysis near the position of the new reference liquid level is performed for the time interval using the eigenmode of the liquid surrounded by the reference liquid level and the tank wall surface. The initial liquid level is obtained by repeatedly repeating the above steps each time the displacement from the reference liquid level reaches 1/10 of the radial dimension of the tank, and obtaining the liquid level position at the end of the time interval. Since it is a method for predicting the behavior of large amplitude sloshing from the reference position , the following excellent effects are exhibited.
(1) Even sloshing the liquid in the tank liquid surface displacement from the initial position increases occur, and sequentially updates the liquid level reference position, of the updated liquid level reference position Vibration Analysis of near It can be solved by repetition.
(2) The updated new reference liquid surface, a spherical coordinate suitable for performing the analysis of the vibration can be easily set. For this reason, the characteristic function of the liquid motion is analytically determined using the newly set spherical coordinates, and the analysis of the liquid motion until the displacement from the new reference liquid surface becomes large by modal analysis is repeated. Therefore, it is possible to apply an analytical method to the problem of sloshing with a large amplitude that has been solved only by CFD. Furthermore, by applying such an analytical method, the degree of freedom can be greatly reduced, and the calculation time and cost required for predicting the behavior of a large amplitude sloshing can be greatly reduced.
(3) By updating the position of the reference liquid level every time the displacement of the liquid level in the tank reaches 1/10 of the radial dimension, the reference liquid level is updated each time the reference liquid level is updated. Based on the measured reference liquid level, vibration analysis near the liquid level reference position may be performed within a range of about 1/10 of the radial dimension. Therefore, the behavior of large amplitude sloshing is predicted by repeating minute amplitude vibration analysis. It becomes possible to do.

  The best mode for carrying out the present invention will be described below with reference to the drawings.

  FIG. 1 shows an embodiment of the large amplitude sloshing behavior prediction method of the present invention, which is as follows.

  Here, first, a method for extending the analysis method proposed in Non-Patent Documents 1, 2, and 3 to the analysis of the behavior of large amplitude sloshing in the present invention will be described.

In FIG. 1, O ₀ XYZ is a coordinate system fixed to the tank 1, and the tank 1 has an axisymmetric shape with respect to the Z axis. The liquid is initially in a static equilibrium state under gravity in the -Z direction g (t) =-e _Z g _init (t <0) (e _Z is a unit vector in the Z direction), and the gravity is g (t) _{= e X g X + e Z} g Z liquid 2 when changed to (t ≧ 0) is considered a large amplitude sloshing problem largely moved from the initial position.

The vibration analysis near the initial position can be performed by the analysis methods described in Non-Patent Documents 1, 2, and 3. In the present invention, in order to expand and develop this analysis method to a sloshing problem with a large amplitude, the liquid level displacement has grown (increased) on the reference liquid level (the liquid level serving as a base point for measuring the liquid level displacement). Each time the value is reached, it is updated near the liquid level at that time, and is described in Non-Patent Documents 1, 2, and 3 in the analysis of each time interval t _n <t <t _{n + 1} until the reference liquid level is updated. Considered to apply the analysis method.

  By the way, when the reference liquid level is updated as described above, the new reference liquid level formed by the update is not axisymmetric with respect to the Z axis. The problem is that there is no longer a cone that contacts the tank wall at the line of contact. However, in the analysis methods described in Non-Patent Documents 1, 2, and 3, as described above, the use of a cone that contacts the tank wall surface at the contact intersection line between the reference liquid surface and the tank wall surface is a key to the analysis method. It was a technique.

ここで、ζは液面の基準液面からの変位である。 In order to solve this problem, in the present invention, as described above, every time the liquid level displacement grows (increases) and reaches a certain value, for example, about 1/10 of the radial dimension, the liquid at that time. When the reference liquid level is updated near the surface, first, the principal axis of inertia of the updated reference liquid level is obtained, and the one that is substantially orthogonal to the liquid level is defined as the z-axis (see FIG. 1). Next, orthogonal coordinates x, y, and z are set so that the x-axis is included in the XZ plane with the intersection of the z-axis and the tank wall surface as the origin, and the divided section ψ of the circumferential angular coordinate ψ around the z-axis Contact intersection line of the reference liquid surface with the tank wall surface every _j ≦ ψ ≦ ψ _{j + 1} (ψ _j = 0.5Δψ + (j−1) Δψ, Δψ = 2π / N, j = 1, 2,..., N) Then, a tangent line is drawn from the intersection of the plane ψ = ψ _j to the tank wall surface, and the spherical coordinate ORθψ is taken with the intersection of the tangent and the z axis as the origin. Such a spherical coordinate ORθψ is illustrated in FIG. 1 for divided section numbers j = j ₁ , j ₂ . Using this spherical coordinate ORθψ, the reference liquid surface M, the liquid surface F displaced from the reference liquid surface, and the tank wall surface W are expressed as follows.

Here, ζ is the displacement of the liquid level from the reference liquid level.

  In the above, since the liquid level is not axially symmetric with respect to the z axis, the z coordinate of the origin changes from positive to negative at a certain circumferential coordinate, and the liquid level displacement ζ taken in the plus R direction can be discontinuous. Is concerned. In order to avoid this problem, a constant ε set to +1 and −1 is introduced depending on whether the origin is on the + z side or the −z side with respect to the tank, and the liquid level is directed outward rather than ζ. Consideration is given to develop and display the displacement -εζ as a characteristic function (see equation (27) described later).

For the convenience of the following analysis, the relationship between the motion coordinate system x≡ (x, y, z) ^t and the tank fixed coordinate system X≡ (X, Y, Z) ^t is expressed as follows.

このような球座標ＯＲθψによる解析は、外側に凸の任意のタンク壁面Ｗ形状に対して特性関数の解析的決定ができ、計算時間、コストが低減できる以外に、以下の利点を有する。 Therefore, the relationship between the spherical coordinate ORθψ and the motion coordinate system x≡ (x, y, z) ^t is given by the following equation.

Such analysis based on the spherical coordinate ORθψ has the following advantages in addition to the ability to analytically determine the characteristic function for an arbitrary outwardly protruding tank wall surface W shape, and to reduce calculation time and cost.

That is, first, even if the liquid level is strongly curved due to surface tension in a low gravity field, the liquid level can be expressed by a monovalent function. Secondly, for any tank wall surface shape, a conforming condition that the liquid level displacement contacts the wall surface at the tank wall surface W is satisfied.
Next, the liquid motion analysis for each time section will be described under the condition where the spherical coordinate ORθψ is set as described above.

Here, first, a variational principle equivalent to the above governing equation system is derived. When there is no surface tension, the Lagrangian density is equal to the liquid pressure. When the surface tension is important under low gravity, the potential energy due to the surface energy at the atmospheric pressure and the gas-liquid interface, the solid-liquid interface, or the solid-gas interface. Therefore, the variational principle is as follows.

The hydraulic pressure p ₁ is expressed as follows by using a velocity potential φ representing a liquid motion relative to the tank, from a pressure equation relating to an unsteady flow.

ここで支配方程式系を表す部分を次のようにおいている。

Substituting Equation (7) into Equation (6) and taking variations with respect to φ, ζ, and G, the following is obtained (refer to Non-Patent Document 1 and Non-Patent Document 2 for detailed derivation methods).

Here, the part representing the governing equation system is as follows.

From the arbitrary independence of the above-described variations of φ, ζ, and G, E _i = 0 (i = 1, 2,..., 5) is a governing equation that should hold. Their physical meaning is as follows.
E ₁ = 0 represents a continuous condition in the liquid region, and becomes a Laplace equation by assuming a vortexless flow.
E ₂ = 0 represents a boundary condition where the flow velocity component in the normal direction is 0 on the tank wall surface W assumed to be a rigid body.
E ₃ = 0 represents a boundary condition that the liquid surface and the velocity component in the liquid surface normal direction of the fluid particles are equal.
E ₄ = 0 represents a force balance condition among atmospheric pressure, liquid pressure, and surface tension, and indicates that a gas-liquid interface pressure difference occurs when surface tension becomes important in a low-gravity universe.
E ₅ = 0 is a condition representing that the contact angle takes a constant value corresponding to the three interfacial tensions during the liquid surface motion.
E ₆ = 0 is a constant volume condition based on the assumption of incompressibility of the liquid.

Since the constant volume condition can be derived from other kinematic conditions E ₁ = E ₂ = E ₃ = 0, E _i = 0 (1, 2,..., 5) can be regarded as a basic expression.

  Next, the spherical coordinate display based on the variational principle will be described.

ここで

上記式（１９）で、Ｓ_Ｍ ^（０）等は液面の基準位置Ｒ_Ｍの関数である。 N _F , N _W , dF, dW, dC, cos θ ′ _C in the above equation (8) are expressed by spherical coordinates ORθψ based on the differential geometry from the equations (2) and (3), and the terms of the liquid surface boundary condition Then, if the radial coordinate (formula (2)) of the liquid level is substituted and the Taylor expansion is performed for the liquid level displacement ζ and linearized, the following is obtained.

here

In the above formula ^{_{(19), S M (0}} ) and the like is a function of the reference position _{R M} of the liquid surface.

と表し、＾ｇ_Ｘ及び＾ｇ_Ｚを、式（２１）の両辺のＸ，Ｚ成分が等しい条件より

と定めて、与えられた重力場を乱さないための外乱として扱う。又、上記式（２１）の変換に応じて、式（１９）の重力項を次のように変換する。

The motion analysis near the reference position of the liquid level is performed using the eigenmode of the liquid surrounded by the reference liquid level M and the tank wall surface W. At this time, in order to prevent the fundamental eigenmode from changing drastically due to a sudden change in gravity, a gravity g _eiz > 0 in the −z direction used for determining the eigenvalue is introduced so as not to disturb the given gravitational field. The residual ^ E ₄ of the mechanical boundary condition at the liquid level is corrected (in this specification, for the sake of convenience, when a letter with a hat (^) in the formula is written in the sentence, the hat (^) ) Before the letter, and so on. That is, given gravity

^ G _X and ^ g _Z are expressed under the condition that the X and Z components on both sides of Equation (21) are equal.

And treat it as a disturbance not to disturb the given gravitational field. Further, according to the conversion of the equation (21), the gravity term of the equation (19) is converted as follows.

の形に基準液面位置の関数Ｓ_Ｍ ^（ｉ）（ｉ＝１０−１５）と液面変位によって表される。したがって、力学的境界条件の残差式（１９）は次のように修正される。

The value on the liquid surface of the orthogonal coordinate in the above equation (23) is obtained by substituting the equation (2) into the equation (5) and converting in the equation (4).

Is expressed by the function S _M ⁽ⁱ⁾ (i = 10-15) of the reference liquid level position and the liquid level displacement. Therefore, the residual equation (19) of the dynamic boundary condition is corrected as follows.

  Next, discretization by the Galerkin method will be described.

上記において、特性関数Ｘ_ｍｋｌ、Θ_ｍｋ等の導出法は非特許文献１及び非特許文献２に記してあるので、ここでは省略する。ここで、特に注意すべきことは、緯角の範囲が０≦θ≦πである通常の球座標では、Θ_ｍｋ（θ）は陪ルジャンドル多項式であるが、ここで用いる球座標は０≦θ≦θ_ｍａｘ＜π／２であるため（図１参照）、Θ_ｍｋ（θ）が無限級数となることである。従ってΘ_ｍｋ（θ）を新たに導く必要がある。 By using the discretization method based on the Galerkin method for the variation principle derived as described above, an ordinary differential equation related to time that governs the liquid surface behavior in each time interval is derived. This discretization method is a method of deriving an ordinary differential equation related to an expansion coefficient (an unknown time function and called generalized coordinates) by expressing the solution in a form expanded by a characteristic function and substituting it into a variational principle. The expansion formula of the solution can be expressed in the following form.

In the above description, the derivation methods for the characteristic functions X _mkl , Θ _{mk and the} like are described in Non-Patent Document 1 and Non-Patent Document 2, and will be omitted here. Here, it should be particularly noted that Θ _mk (θ) is a Legendre polynomial in normal spherical coordinates where the range of latitude is 0 ≦ θ ≦ π, but the spherical coordinates used here are 0 ≦ θ. Since ≦ θ _max <π / 2 (see FIG. 1), Θ _mk (θ) is an infinite series. Therefore, it is necessary to newly derive Θ _mk (θ).

ここでａ，ｃは、それぞれ一般化座標ａ_ｍｋｌｑ，ｃ_ｍｋｑを並べた列ベクトルである。上記式（２８）のａ，ｃに関する方程式からａを消去すると次のようになる。

ここで

Substituting Equation (26) and Equation (27) into Equation (15) of the variation principle and taking variations for the generalized coordinates a _mklq and c _mkq , the following matrix equation is derived.

Here, a and c are column vectors in which generalized coordinates a _mklq and c _mkq are arranged, respectively. When a is eliminated from the equations relating to a and c in the above equation (28), the result is as follows.

here

となる。 When a matrix T in which each column is an eigenvector of the matrix M ⁻¹ K ₂₂ is used, and conversion c = Tq into mode coordinates q,

It becomes.

The coefficient of q on the left side of the above equation (31) is a diagonal matrix having the square of the natural frequency as a diagonal element. By solving the above equation (31), the liquid level position R _n = R _{F, n} (θ _n , ψ _n , t _{n + 1} ) at the end time of the n-th time interval of interest is determined.

  The feature of this analysis method is that the analysis within the above time interval can be performed at a higher speed and at a lower cost than the conventional numerical methods such as CFD due to the following two points.

  That is, first, the characteristic function is analytically determined by introducing the spherical coordinate ORθψ. Second, the dimension of the eigenvalue problem for mode conversion can be small.

As described above, the principal axis of inertia of the liquid level at the end time of the time interval becomes the z axis in the next time interval, that is, z _{n + 1} , which is substantially orthogonal to the liquid level. Accordingly, an orthogonal coordinate system (x _{n + 1} , y _{n + 1} , z _{n + 1} ) is defined with the intersection of the z _{n + 1} axis and the tank wall surface W as the origin, and y _{n + 1} in the XZ plane, and spherical coordinates (R _{n + 1} , θ as follows) _{n + 1} , ψ _{n + 1} ).

ここで、Ｈは初期液面のｚ_ｎ＋１軸との交点のｚ_ｎ＋１座標であり、Ａ，Ｂはこの楕円体面が初期液面の接触線でタンク壁面と接触角θ´_Ｃで交わる条件により周方向各分割区間に対して定める。 This new spherical coordinate (R _{n + 1} , θ _{n + 1} , ψ _{n + 1} ) is obtained by converting the liquid surface position R _n = R _Fn (θ _n , ψ _n , t _{n + 1} ) at the end time of the previous time interval R _{n + 1} = R _{F, n + 1} (θ _{n + 1} , ψ _{n + 1} , t _{n + 1} ) is determined as the initial liquid level of the new time interval. In order to obtain the initial value of the liquid level displacement, first, the reference liquid level in the new time interval is set as the next ellipsoidal plane.

Here, H is the z _{n + 1} coordinate of the intersection with the z _{n + 1} axis of the initial liquid level, and A and B are the circumferences depending on the condition that the ellipsoidal plane intersects the tank wall surface at the contact angle θ ′ _C with the contact line of the initial liquid level. The direction is determined for each divided section.

Substituting equation (32) into equation (33) and solving for R _{n + 1} determines the reference liquid level R _{n + 1} = R _{M, n + 1} (θ _{n + 1} , ψ _{n + 1} ). The outward displacement −ε (R _{F, n + 1} −R _{M, n + 1} ) of the initial liquid level from the reference liquid level determined in this way is expressed by the following equation on the basis of the equation (27) representing the liquid level displacement allowable function. When the Fourier expansion is performed as shown above, a simultaneous linear equation related to the initial value c _mkq (t _{n + 1} ) of the generalized coordinates is obtained, and the initial value c _mkq (t _{n + 1} ) is determined by solving this.

Therefore, according to the large amplitude sloshing behavior prediction method of the present invention having the above configuration, even if sloshing occurs in the liquid in the tank and the liquid level displacement from the initial position increases, the above equation (31) is solved. Therefore, the liquid level position R _n = R _{F, n} (θ _n , ψ _n , t _{n + 1} ) at the end time of the n-th time interval of interest is determined by Since the initial static equilibrium position can be updated sequentially near the current liquid level, the updated liquid level reference position can be set as the static equilibrium position and the vibration can be analyzed. The sloshing problem can be solved by repeating the vibration analysis near the static equilibrium position.

  Specifically, among the inertial principal axes of the new reference liquid surface, the one substantially orthogonal to the liquid surface is taken as the new z axis, and the circumferential angle coordinate ψ is taken around the z axis, and this circumferential angle coordinate ψ is Dividing into equal parts, a tangent line is drawn from the contact intersection line of the new reference liquid surface to the tank wall surface in each divided section, and the spherical coordinates are set with the intersection point of the tangent line and the z axis as a new origin. Therefore, the characteristic function of the liquid motion is analytically determined using the spherical coordinates, and the liquid motion until the displacement from the new reference liquid surface becomes large is analyzed by the mode analysis. By repeating this, it becomes possible to apply an analytical method to the problem of sloshing with a large amplitude that has been solved only by CFD. Furthermore, by applying such an analytical method, the degree of freedom can be reduced. In the CFD, the matrix dimension is several hundred to thousands, whereas the large-amplitude sloshing behavior of the present invention. According to the prediction method, the matrix can be significantly reduced to about five dimensions. Therefore, the calculation time and cost required for predicting the behavior of sloshing with large amplitude can be greatly reduced.

  The present invention is not limited to the above embodiment, and the reference liquid level that is updated when a large amplitude sloshing that causes the liquid 2 in the tank 1 to move in a wide range in the tank 1 occurs. By drawing the tangent line with the tank wall surface at the contact intersection line with the tank wall surface, it is possible to determine the intersection point between the tangent line and the new z-axis substantially orthogonal to the updated reference liquid surface. It can be applied to predicting the behavior of large amplitude sloshing in any tank shape. Of course, various modifications can be made without departing from the scope of the present invention.

  Hereinafter, the result of verification of the effectiveness of the present invention performed by the present inventor will be described.

又、重力の大きさは無次元のボンド数で与えている。重力方向は初期において−Ｚ方向で、初期時刻にＸ軸から−２０度の方向に変化した場合を計算した。液体量は、初期において上に指定したボンド数と接触角の下で、接触線のＺ座標が０．３９ｍとなる液体量としてある。 In the configuration of FIG. 1, the calculation was performed under the following conditions with the tank having a spherical shape with a radius of b = 0.3 m.

The magnitude of gravity is given by the dimensionless bond number. The gravity direction was initially -Z direction, and the case where it changed from the X axis to -20 degrees at the initial time was calculated. The liquid amount is such that the Z coordinate of the contact line is 0.39 m under the bond number and contact angle specified above in the initial stage.

  2A and 2B are cross-sectional views of the liquid on the plane of ψ = 15 degrees to 225 degrees. The broken line in the figure indicates the static equilibrium position of the liquid in the initial stage.

  As shown in FIG. 2 (a), it was found that the liquid reached the final position as shown in FIG. 2 (b) after overshooting over the final position at an intermediate time (t = 70 sec). . The final position is symmetric about a straight line that passes through the center of the tank parallel to the direction of gravity after fluctuation. FIG. 3 shows the time transition of the X and Z coordinates at a point of ψ = 15 degrees on the contact line between the liquid level and the tank wall surface. This point converges near the intersection of the tank wall surface with the Z-axis, and the tank wall surface is almost perpendicular to the Z-axis in this vicinity, so it turns out that the vibration around the final value of the Z coordinate at this point is very small. did.

It is a schematic diagram which shows one Embodiment of the large amplitude sloshing behavior analysis method of this invention. The results of verification of the effectiveness of the present invention conducted by the present inventors are shown. (A) shows the state in which the liquid in the tank is most displaced from the initial static equilibrium position, and (B) shows the state in the tank. It is sectional drawing of the liquid which shows the last position of a liquid, respectively. FIG. 3 is a diagram showing a time transition of X and Z coordinates of a point of a required position on a contact line between a liquid surface and a tank wall surface as a verification result of FIG. 2. It is a schematic diagram which shows the method for the analysis of sloshing in case the liquid shape at the time of the static equilibrium which this inventor proposes becomes axisymmetric.

Explanation of symbols

1 Tank 2 Liquid 3 Static equilibrium liquid level 4 Contact line

Claims (1)

Using a natural mode of the liquid surrounded by the reference liquid surface and the tank wall surface, the computer analyzes the motion near the reference position of the liquid surface using the initial position where the liquid in the tank is in a static equilibrium state as the reference position. When the displacement of the liquid level in the tank from the reference liquid level reaches 1/10 of the radial dimension of the tank , the liquid level after the liquid level displacement at that time is changed to the position of the new reference liquid level. And a step of obtaining a time interval until the reference position of the initial liquid level is updated to the position of the new reference liquid level, a step of obtaining the inertia main axis of the updated new reference liquid level, and Taking the inertial main axis orthogonal to the updated reference liquid surface as an axis, taking the circumferential angle coordinate around the axis, equally dividing the circumferential angle coordinate, and after the update in each divided section From the contact intersection line of the reference liquid level with the tank wall surface to the tank wall surface Draw a line, a step of determining the spherical coordinates of the liquid surface after the liquid surface displacement of the intersection of the該接line and the axis as a new starting point, under the conditions described above spherical coordinate has been determined for said time interval, The liquid surface motion analysis near the position of the new reference liquid surface is performed using the eigenmode of the liquid surrounded by the reference liquid surface and the tank wall surface, and the displacement from the reference liquid surface is The above process is repeated in sequence every time the radius size reaches 1/10, and the behavior of large amplitude sloshing from the initial liquid level reference position is predicted by obtaining the liquid level position at the end of the time interval. A large amplitude sloshing behavior prediction method.

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