Review of the Dillingham, Falzarano & Pantazopoulos rotating three-dimensional shallow-water equations

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1 Review of the Dillingham, Falzarano & Pantazopoulos rotating three-dimensional shallow-water equations by H. Alemi Ardakani & T. J. Bridges Department of Mathematics, University of Surrey, Guildford GU2 7XH UK November 22, Introduction Two derivations of the shallow water equations (SWEs) for fluid in a vessel that is undergoing a general rigid-body motion in three dimensions first appeared in the literature at about the same time, given independently by Pantazopoulos [8, 9, 10] Dillingham & Falzarano [3]. Both derivations follow the same strategy. Their respective derivations are an extension of the formulation for two-dimensional shallow water flow in a rotating frame in [2]. Their idea is to start with the classical SWEs u t + uu x + vu y + gh x = 0 v t + uv x + vv y + gh y = 0 h t + (hu) x + (hv) y = 0, (1.1) where h(x, y, t) is the free surface elevation, u, v are representative horizontal velocities g > 0 is the gravitational constant. They then deduce the acceleration a (x) := (a (x), a (y), a (z) ), of the body frame relative to an inertial frame. Then g is replaced by an average of a (z) approximations for a (x) a (y) are substituted into the right-h side of the horizontal momentum equations. The resulting SWEs will be called the DFP SWEs. These equations were also later used in [4] to study the dynamical behaviour of an offshore supply vessel with water on deck. The purpose of this report is to determine the precise approximations used in the derivation in order to compare with the new shallow-water equations found in [1]. 1

2 2 Summary of DFP SWEs Pantazopoulos [8, 9, 10] Dillingham & Falzarano [3] propose the following form for the 3D rotating shallow-water equations u t + uu x + vu y + a (z) h x = f 1 v t + uv x + vv y + a (z) h y = f 2 h t + (hu) x + (hv) y = 0, (2.2) where f 1 f 2 are representations of the horizontal accelerations due to the vessel motion. In the sequel we will use [10] as a guide to the derivation as it gives the most detail. The final equations in all the sources are the same modulo typographical errors. The expression given for f 1 is f 1 = n 1 cosθ n 2 sin φ sin θ + n 3 sin θ cos φ 2ω 1 ω 2 x sin φ sinθ cosθ ω 1 ω 2 y cosφcosθ ω 1 ω 2 z d sin φ(sin 2 θ cos 2 θ) + ω 2 1 x sin2 θ ω 2 1z d sin θ cosθ + ω 2 2x(1 sin 2 θ sin 2 φ) ω 2 2y sin φ sinθ cosφ +ω 2 2 z d sin θ sin 2 φ cosθ + 2ω 1 v sin θ 2ω 2 v sin φ cosθ + ω 1 y sin θ ω 2 y sin φ cosθ ω 2 z d cosφ + g sin θ cosφ, (2.3) after correcting typos adding in a missing term (boxed in the above equation). This missing term also appears to be a typo since the boxed term is included in a (x) in [10] f 1 = a (x) + u. The expression for f 2 in [10] is f 2 = n 2 cosφ n 3 sin φ + ω 2 1 y ω2 2 x sin φ sin θ cosφ + ω2 2 y sin2 φ +ω 2 2z d sin φ cosφcosθ ω 1 ω 2 x cosφcosθ ω 1 ω 2 z d sin θ cosφ 2ω 1 u sin θ + 2ω 2 u sin φ cosθ ω 1 x sin θ + ω 1 z d cosθ + ω 2 x sin φ cosθ + ω 2 z d sin θ sin φ g sin φ, after correction of two typos. The vertical acceleration term is a (z) = n 1 sin θ n 2 sin φ cosθ + n 3 cosφcosθ + ω 1 ω 2 x sin φ(sin 2 θ cos 2 θ) +ω 1 ω 2 y sin θ cosφ 2ω 1 ω 2 z d sin φ sin θ cosθ +ω 2 1 x sin θ cosθ ω2 1 z d cos 2 θ ω 2 2 x sin θ sin2 φ cosθ ω 2 2y sin φ cosφcosθ ω 2 2 z d (1 sin 2 φ cos 2 θ) + 2ω 1 v cosθ +2ω 2 v sin φ sinθ 2ω 2 u cosφ + ω 1 y cos θ ω 2 x cosφ (2.4) + ω 2 y sin φ sin θ + g cosφcosθ. (2.5) In the term ω2 2 z d (1 sin 2 φ cos 2 θ) in [8, 10], the boxed part, z d, is missing it has been reinstated here. The same equations are used in [3, 4, 9]. We have not seen reference [3] but [4] imply that the equations are the same as above. 2

3 3 Major assumption on the angular velocity In these equations ω = (ω 1, ω 2, ω 3 ) is the spatial angular velocity the reader will note that ω 3 does not appear anywhere in the above expressions. This missing term is because of the following assumption ω 3 = 0. (DFP-1) This assumption is surprising. It is both inconsistent unnecessary. It is inconsistent because the third component of the body angular velocity is not set to zero as well (see below). It is unnecessary because the authors develop these equations for numerical simulation so inclusion of ω 3 is straightforward. The use of spatial angular velocity rather than body angular velocity is also what causes the expressions for f 1, f 2 the vertical acceleration to be excessively complicated. 4 Moving frame of reference Let X = (X, Y, Z) be coordinates for a fixed spatial frame let x = (x, y, z) be coordinates for the body frame. Then DFP use the following relationship between the fixed frame the moving frame X = Q(x + d) + n, with n = (n 1, n 2, n 3 ) the spatial translation of the moving frame. The following assumptions are imposed on x d x x = y, (DFP-2) 0 0 d = 0. z d The rotation matrix is restricted to cosθ 0 sin θ Q = sin φ sin θ cos φ sin φ cosθ. cosφsin θ sin φ cosφcosθ (DFP-3) (DFP-4) This matrix is given explicitly in equation (6) on page 29 of [10]. The angles φ θ are Euler angles associated with roll pitch respectively. A derivation of (DFP-4) using rotation tensors is given in the next section. 3

4 z Z θ, pitch y Y φ, roll X, x Figure 1: Schematic of the roll-pitch motion in terms of Euler angles φ θ. 5 Prescribed roll-pitch motion of the vessel A derivation of the rotation matrix (DFP-4) is given in terms of rotation tensors. Let {E 1,E 2,E 3 } be a basis for the spatial frame X. Using a coordinate-free rotation tensor formulation [7], the roll motion is where R := L(φ,E 1 ) = cosφ(i E 1 E 1 ) + sin φê1 + E 1 E 1, (5.1) The hat-matrix has the property that (a b)c := (b c)a, a,b,c R 3, 0 a 3 a 2 â := a 3 0 a 1. (5.2) a 2 a 1 0 âb = a b, a,b R 3. The rotation (5.1) is a counterclockwise rotation about the X axis as shown schematically in Figure 1. The matrix representation of the rotation (5.1) is R = 0 cos φ sin φ. 0 sin φ cosφ 4

5 However it is not a good idea to use a matrix representation until the full rotation matrix is constructed since the basis for the rotation is changing. The pitch rotation is a counterclockwise rotation about the y axis. The axis of rotation is The pitch rotation is therefore a 2 := RE 2 = cosφe 2 + sin φe 3. P := L(θ,a 2 ) = cosθ (I a 2 a 2 ) + sin θâ 2 + a 2 a 2. The roll-pitch rotation is then Q = PR. In order to construct a matrix representation of this composite rotation, use the property of rotation tensors L(θ,Rb) = RL(θ,b)R T. when R is any proper rotation (see [5, 6] for a proof of this identity). Applying this property to Q gives Q = PR = L(θ,RE 2 )L(φ,E 1 ) = RL(θ,E 2 )R T R = L(φ,E 1 )L(θ,E 2 ). Since both matrices are now relative to the stard basis, a matrix representation can be constructed, cosθ 0 sin θ Q = 0 cosφ sin φ 0 1 0, 0 sin φ cos φ sin θ 0 cosθ which when multiplied out gives (DFP-4). The angular velocities are now easily computed. Considering φ θ as functions of time 0 Ω s QQ T 3 Ω s 2 = Ω s 3 0 Ω s 1 = Ω s, Ω s 2 Ω s 1 0 where Ω s = (Ω s 1, Ω s 2, Ω s 3) is the spatial angular velocity vector. Now d dt L(γ,E j)l(γ,e j ) T = γêj, Q = L(φ,E 1 )L(θ,E 2 ) + L(φ,E 1 ) L(θ,E 2 ) = φê1q + θl(φ,e 1 )Ê2L(θ,E 2 ) 5

6 so Ω s = QQ T = φê1 + θl(φ,e 1 )Ê2L(φ,E 1 ) T. Now use the identity Qv = Q vq T, for any v R 3, any Q SO(3), to arrive at Ω s = φe 1 + θl(φ,e 1 )E 2 or in components = φe 1 + θa 2 = φe 1 + θ(cosφe 2 + sin φe 3 ), φ Ω s = θ cosφ. θ sin φ The body angular velocity spatial angular velocity are related by φcosθ Ω b = Q T Ω s = θ φsin θ. If we assume that θ is small then φ Ω s θ, φ θ φ Ω b θ. θ φ If we further neglect the quadratic terms φ θ θ φ then the angular velocities reduce to φ Ω s = Ω b = θ. 0 This is the approximation used explicitly by [4] appears to be implicitly used in [8, 10]. Formalizing this assumption φ 0 θ 0. (DFP-5) The assumption φ θ 0 implies assumption (DFP-1). However there is an inconsistency in this assumption in that φθ is not assumed to be small! Hence Ω s 3 is assumed to be small but Ω b 3 is not assumed to be small. On the other h since the aim is to numerically simulate, neither assumption (DFP-1) or assumption (DFP-5) is necessary. 6

7 6 Simplifying f 1 f 2 The authors [4, 8, 9, 10] all use the spatial angular velocity in their derivation. But the equations are relative to the body frame. It is certainly correct, but it leads to cumbersome equations. In this section it is shown that the terms f 1 f 2 simplify considerably when the body angular velocity is used. To simplify notation use ω = (ω 1, ω 2, ω 3 ), for the spatial angular velocity as in [8, 10] use Ω = (Ω 1, Ω 2, Ω 3 ), for the body angular velocity as in [1]. With the assumption (DFP-1), the body angular velocity can be expressed in terms of the spatial angular velocity using ω = QΩ, Ω 1 Ω 2 = ω 1 cosθ + ω 2 sin φ sin θ = ω 2 cosφ Ω 3 = ω 1 sin θ ω 2 sin φ cosθ. The translation gravity terms can be expressed using Q, n Qe 1 = n 1 cosθ + n 2 sin φ sin θ n 3 cosφsin θ, ge 3 Qe 1 = g cosφsin θ. (6.3) Using these two expressions, (6.3), the formulae in Appendix A, f 1 simplifies to f 1 = n Qe 1 + (Ω Ω2 3 )x Ω 1Ω 2 y Ω 1 Ω 3 z d +2Ω 3 v Ω 2 z d + Ω (6.4) 3 y ge 3 Qe 1. A similar construction shows that f 2 = n Qe 2 + (Ω Ω 2 3)y Ω 1 Ω 2 x Ω 2 Ω 3 z d 2Ω 3 u + Ω 1 z d Ω 3 x ge 3 Qe 2. (6.5) The simplification in the expressions (6.4) (6.5) over the original expressions (2.3) (2.4) is remarkable. The simplification is due first to the use of the body angular velocity secondly to the explicit use of the rotation operator. To see that these are equivalent to (2.3) (2.4) substitute (6.3) into (6.4) to recover (2.3) substitute (6.3) into (6.5) to recover (2.4). The vertical acceleration term can also be simplified to a (z) = n Qe 3 (Ω Ω 2 2)z d + Ω 1 Ω 3 x + Ω 2 Ω 3 y +2Ω 1 v 2Ω 2 u + Ω 1 y Ω 2 x + ge 3 Qe 3. (6.6) That this expression is equivalent to (2.5) is verified by substitution of (6.3) into (6.6). 7

8 7 Comparison of the surface equations [1] with the DFP SWEs With the coefficients now expressed in terms of the body angular velocity we can compare the DFP SWEs with the new surface equations derived in [1]. The surface momentum equations derived in [1], neglecting surface tension are with U t + UU x + V U y + a 11 h x + a 12 h y = b 1, V t + UV x + V V y + a 21 h x + a 22 h y = b 2, a 11 = 2Ω 1 V + Qe 3 q + gqe 3 e 3 (Ω Ω2 2 )(h + d 3) ( Ω 2 Ω 1 Ω 3 )(x + d 1 ) + ( Ω 1 + Ω 3 Ω 2 )(y + d 2 ) a 22 = 2Ω 2 U (Ω Ω2 2 )(h + d 3) + Qe 3 q + gqe 3 e 3 +( Ω 1 + Ω 2 Ω 3 )(y + d 2 ) ( Ω 2 Ω 1 Ω 3 )(x + d 1 ). a 12 = 2Ω 2 V, a 21 = 2Ω 1 U b 1 = 2Ω 2 h t + 2Ω 3 V Qe 1 q gqe 1 e 3 + (Ω Ω2 3 )(x + d 1) +( Ω 3 Ω 1 Ω 2 )(y + d 2 ) ( Ω 2 + Ω 1 Ω 3 )(h + d 3 ), b 2 = 2Ω 1 h t 2Ω 3 U Qe 2 q gqe 2 e 3 + (Ω Ω 2 3)(y + d 2 ) ( Ω 3 + Ω 1 Ω 2 )(x + d 1 ) + ( Ω 1 Ω 2 Ω 3 )(h + d 3 ). (7.1) (7.2) (7.3) (7.4) The advantage of these equations is that the vessel motion is exact with the only assumption being neglect of the vertical acceleration at the free surface [1]. The velocity field (U, V ) is not the same as the velocity field in the DFP SWEs. Assume for purposes of comparison that (U, V ) (u, v), the the coefficients in the two systems can be compared. First note that q = n, ω 3 = 0 in the DFP equations d 3 = z d. Comparing, a 11 = a (z) + 2Ω 2 U (Ω Ω2 2 )h ( Ω 2 Ω 1 Ω 3 )d 1 + ( Ω 1 + Ω 2 Ω 3 )d 2 a 12 a 21 = a DFP Ω 2 V = a DFP 21 2Ω 1 U a 22 = a (z) 2Ω 1 V (Ω Ω2 2 )h ( Ω 2 Ω 1 Ω 3 )d 1 + ( Ω 1 + Ω 2 Ω 3 )d 2. In the DFP formulation, the coefficients a 12 a 21 are identically zero. The right-h side coefficient comparison is b 1 = f 1 2Ω 2 h t + (Ω Ω2 3 )d 1 + ( Ω 3 Ω 1 Ω 2 )d 2 ( Ω 2 + Ω 1 Ω 3 )h b 2 = f 2 + 2Ω 1 h t + (Ω Ω2 3 )d 2 ( Ω 3 + Ω 1 Ω 2 )d 1 + ( Ω 1 Ω 2 Ω 3 )h 8

9 If we invoke the assumption (DFP-3) then these relations simplify to a 11 = a (z) + 2Ω 2 U (Ω Ω2 2 )h a 12 a 21 = a DFP Ω 2 V = a DFP 21 2Ω 1 U a 22 = a (z) 2Ω 1 V (Ω Ω2 2 )h b 1 = f 1 2Ω 2 h t ( Ω 2 + Ω 1 Ω 3 )h b 2 = f 2 + 2Ω 1 h t + ( Ω 1 Ω 2 Ω 3 )h. Therefore in order to reduce the surface equations to the DFP equations we need the further assumptions 2Ω 2 U (Ω Ω 2 2)h 0 2Ω 1 V + (Ω Ω2 2 )h 0 2Ω 2 h t + ( Ω 2 + Ω 1 Ω 3 )h 0 2Ω 1 h t + ( Ω 1 Ω 2 Ω 3 )h 0 Ω 2 V h y 0 Ω 1 Uh x 0. (DFP-6) With these additional assumptions, equivalence of the velocity fields, the equations reduce to the same form. These assumptions are however quite severe are not justified in general. Appendix A Vector identities used in transforming spatial coordinates to body coordinates Using the transformation (6.3) 2Ω 3 v 2ω 1 v sin θ + 2ω 2 v sin φ cosθ 2Ω u = 2Ω 3 u 2Ω 1 v 2Ω 2 u = 2ω 1 u sinθ 2ω 2 u sin φ cosθ 2ω 1 v cosθ + 2ω 2 v sin φ sin θ 2ω 2 u cosφ. 9

10 Now let x = (x, y, z d ) where z d is the constant vertical distance from the centre of gravity to the deck. Then Ω 2 z d Ω 3 y Ω x = 1 z d + Ω 3 x Ω 1 y Ω 2 x ω 2 z d cos φ y( ω 1 sin θ ω 2 sin φ cosθ) = z d ( ω 1 cosθ + ω 2 sin φ sin θ) + x( ω 1 sin θ ω 2 sin φ cosθ). y( ω 1 cosθ + ω 2 sin φ sin θ) ω 2 x cosφ Similarly (Ω Ω 2 3)x + Ω 1 Ω 2 y + Ω 1 Ω 3 z d Ω (Ω x) = Ω 1 Ω 2 x (Ω Ω2 3 )y + Ω 2Ω 3 z d. Ω 1 Ω 3 x + Ω 2 Ω 3 y (Ω Ω 2 2)z d Translating into spatial angular velocities Ω (Ω x) 1 = xω1 2 sin2 θ xω2 2(1 sin2 φ sin 2 θ) + 2xω 1 ω 2 sin φ sin θ cosθ +yω 1 ω 2 cos φ cosθ + yω2 2 sin φ cosφsin θ + z dω1 2 sin θ cos θ +z d ω 1 ω 2 sin φ(sin 2 θ cos 2 θ) z d ω2 2 sin 2 φ sin θ cosθ, Ω (Ω x) 2 = xω 1 ω 2 cos θ cos φ + xω 2 2 sin φ cosφ sin θ yω2 1 yω 2 2 sin 2 φ + ω 1 ω 2 z d cosφsin θ ω 2 2z d sin φ cosφcosθ, Ω (Ω x) 3 = xω 2 1 sin θ cosθ xω 1 ω 2 sin φ(cos 2 θ sin 2 θ) xω 2 2 sin 2 φ sin θ cosθ +yω 1 ω 2 cos φ sinθ yω 2 2 sin φ cosφcosθ 2z dω 1 ω 2 sin φ sin θ cosθ z d ω 2 1 cos2 θ z d ω 2 2 sin2 φ sin 2 θ z d ω 2 2 cos2 φ, B Sketch of the derivation of Pantazopoulos [10] In his thesis, Pantazopoulos gives a derivation of the acceleration terms relative to the moving frame. Here a sketch of that derivation is given which shows how the use of the spatial angular velocity complicates the equations. The general relation between X x is X = Q(x + d) + n, (B-1) where Q is a rotation matrix, d is the distance to the centre of rotation n is the displacement of the body frame. In [10] it is assumed that x = (x, y, 0), d = (0, 0, z d ) Q is restricted to the form (DFP-4). To simplify notation let x = (x, y, z d ), (B-2) 10

11 Hence (B-1) simplifies to X = Qx + n, With derivative Ẋ = Qx + Qẋ + ṅ. Use the spatial angular velocity ω defined by (B-3) (B-4) QQ T = ω, where the hat-map is defined in (5.2). Hence (B-4) has the equivalent representation Ẋ = ω (X n) + Qẋ + ṅ. Note also that Ẋ ṅ = ω (Qx) + Qẋ. Differentiate (B-5) again to obtain the absolute acceleration Ẍ = ω (X n) + ω (Ẋ ṅ) + Qẋ + Qẍ + n, (B-5) (B-6) (B-7) or after substitution, Ẍ = n + Qẍ + ω (ω (Qx)) + 2ω (Qẋ) + ω (Qx). (B-8) This latter equation is in fact equation (5) in [10] with the identifications r = n ρ = Qẍ ρ = Qx ρ = Qẋ. [10] makes the assumption ω = (ω 1, ω 2, 0) as noted in (DFP-1). A problem is that the right-h side of (B-8) has a mixture of spatial vectors body vectors: x is the position in the body coordinates, but ω is the spatial angular velocity. [10] proceeds to derive the acceleration relative to the body. Let A s := Ẍ, be the acceleration relative to the fixed frame let A b = Q T A s, (B-9) be the acceleration viewed from the body frame. In [10] this latter vector is denoted by a (x) A b := a (y). a (z) 11

12 [10] proceeds to derive explicit expressions for the components of A b (in equation (10a) (10b) (10c) on page 33 of [10]). However a general expression can be found which highlights why the formulae in [10] are so complicated. Combining (B-8) with (B-9) gives A b = Q T A s = Q T ( n + Qẍ + ω (ω (Qx)) + 2ω (Qẋ) + ω (Qx)). Before analyzing this expression, note that ẋ is the Lagrangian velocity ẍ the Lagrangian acceleration of a fluid particle. Replacing these terms gives ( A b = Q T A s = Q T n + Q Du ) + ω (ω (Qx)) + 2ω (Qu) + ω (Qx). Dt (B-10) To simplify further use the following fundamental identity [7]: for any vectors a, b R 3 any invertible 3 3 matrix M, Ma Mb = det(m) ( M 1)T a b. If M=Q Q is a proper rotation (det(q) = 1 Q 1 = Q T ) then this formula simplifes to Qa Qb = Q(a b). Apply this formula to (B-10) A b = Q T n + Du Dt + QT ω (Q T ω x) + 2Q T ω u + Q T ω x. (B-11) Hence it is clear that if we insist on using the spatial angular velocity then the equation will be very complicated. [10] treats gravity as an acceleration (it is in fact a body force) so A b becomes A b = Q T n + Du Dt + QT ω (Q T ω x) + 2Q T ω u + Q T ω x + gq T e 3, (B-12) where g > 0 is the gravitational constant. It is precisely the components of A b in (B-12) which appear in equation (10) on page 33 of [10]. If instead we replace the spatial angular velocity with the body angular velocity then the equation simplifies dramatically Ω := Q T ω, A b = Q T n + Du Dt + Ω (Ω x) + 2Ω u + Ω x + gq T e 3. (B-13) In general, one can use either (B-12) or (B-13) to write out the components of A b, but the latter expression is less complicated. 12

13 References [1] H. Alemi Ardakani & T.J. Bridges. Shallow-water sloshing in rotating vessels. Part 2. Three-dimensional flow field, Preprint: University of Surrey (2009). [2] J.T. Dillingham. Motion studies of a vessel with water on deck, Marine Technology (1981). [3] J.T. Dillingham & J.M. Falzarano. Three-dimensional numerical simulation of green water on deck, Third International Conference on the Stability of Ships Ocean Vehicles, Gdansk, Pol (1986). [4] J.M. Falzarano, M. Laranjinha & C. Guedes Soares. Analysis of the dynamical behaviour of an offshore supply vessel with water on deck, Proc. 21st Inter. Conf. Offshore Mechanics Arctic Eng. (OMAE02), Paper No. OMAE2002-OFT28177, ASME (2002). [5] C. Leubner. Coordinate-free rotation operator, Amer. J. Phys (1979). [6] C. Leubner. Correcting a widespread error concerning the angular velocity of a rotating rigid body, Amer. J. Phys (1981). [7] O.M. O Reilly. Intermediate Dynamics for Engineers: a Unified Treatment of Newton-Euler Lagrangian Mechanics, Cambridge University Press (2008). [8] M. S. Pantazopoulos. Three-dimensional sloshing of water on decks, Marine Technology (1988). [9] M. S. Pantazopoulos & B. H. Adee. Three-dimensional shallow water waves in an oscillating tank, Proc. Amer. Soc. Civil Eng. ASCE (1987). [10] M. S. Pantazopoulos. Numerical solution of the general shallow water sloshing problem, PhD Thesis: University of Washington, Seattle (1987). 13