Geometric Algebra Approach to Fluid Dynamics

Transcription

1 Geometric Algebra Approach to Flui Dynamics Carsten Cibura an Dietmar Hilenbran Abstract In this work we will use geometric algebra to prove a number of well known theorems central to the fiel of flui ynamics, such as Kelvin s Circulation Theorem an Helmholtz Theorem, showing that it is accessible by geometric algebra methos an that these methos facilitate the representation of an calculation with flui ynamics concepts. Then we propose a generalization of the stream function to arbitrary imensions, extening its explanatory power to higher-imensional flows. We will show how this extene stream function behaves on streamlines an we will relate it to the curl of the flow s vector fiel, i.e. its vorticity. 1 Introuction While geometric algebra methos have been successfully applie to a variety of problems in the recent past, incluing computer vision [10], robotics [3] [8] an relativistic electro-magnetic fiel theory [1] [5], its applications to the fiel of flui ynamics have been restricte to isolate subjects or specialize problem escriptions, e.g. [4] [6] [9]. The systematic evelopment of a comprehensive flui ynamics calculus using geometric algebra methos is still an open task. In this work we will use geometric algebra to prove a number of well known theorems central to the fiel of flui ynamics, showing that some conclusions follow neatly from geometric properties (e.g. of vector fiels) encoe in the calculational rules of the geometric prouct. Then we propose a generalization of the stream Carsten Cibura Technische Universität Darmstat, currently at Universiteit van Amsteram, Faculteit er Natuurwetenschappen, Wiskune en Informatica, Kruislaan 404, 1098 SM Amsteram, c.cibura@uva.nl Dietmar Hilenbran Technische Universität Darmstat, Fraunhoferstr. 5, Darmstat, ietmar.hilenbran@gris.informatik.tu-armstat.e 1

2 2 Carsten Cibura an Dietmar Hilenbran function to arbitrary imensions. Classically, the stream function is efine for twoimensional flows, where it can be use to calculate stream lines or the (complex) potential of a flow. Generalizing the stream function to arbitrary imensions offers extene possibilities of analyzing a flui flow. We will procee as follows. In section 2 we will give a short introuction into what we consier avance concepts of geometric algebra, like vector ifferentiation an integration. Section 3 will introuce the reaer to some pecularities of flui ynamics on a rather algebraic level, to an extent that is neee to unerstan this work. For a more extene an figurative introuction, see [2]. In section 4 we use geometric algebra to prove a number of theorems central to flui ynamics, such as Kelvin s Circulation Theorem an Helmholtz Theorem. Also, we will show how to efine the stream function, which is recognize as useful for the escription of 2D flows, in arbitrary imensions. Finally, in section 5, we will summarize our finings. 2 Geometric Algebra We assume that the reaer is familiar with the basics of geometric algebra, i.e. the inner, outer an geometric prouct, the way these apply to vectors, how they exten to higher grae objects, the role of the pseuoscalar etc. In the following we will give a brief introuction into vector ifferentiation an irecte integration. For a comprehensive an more axiomatic introuction we refer the reaer to [7]. [5] gives an introuction that might be consiere more accessible. 2.1 Vector Differentiation The vector erivative unites the algebraic properties of a vector with those of an operator. This becomes clear, if one expans it in terms of a basis. Given a coorinate frame {e k } an its reciprocal frame {e k }, one can write = k e k e k = k e k x k, (1) which shows that inherits the properties of the partial erivatives i.e. chain rule an prouct rule apply while at the same time it plays the role of a vector in any prouct. The application of the vector erivative to scalar an vector-value functions or fiels, respectively, may serve as an illustration of its calculational power. Let p(x) be a scalar-value fiel, i.e. a function epening on n-imensional position x. Then p(x) = e k p(x) k x k, (2) is the vector-value graient of the scalar fiel p(x).

3 Geometric Algebra Approach to Flui Dynamics 3 Things get more interesting when we apply to a vector-value function, e.g. the vector fiel u(x). The full vector erivative yiels u(x) = u(x) + u(x), (3) a general multivector consisting of two terms of grae zero an two, respectively. The scalar term u(x), which is the ivergence of u(x) an a pure bivector u(x), which is the vector fiel s curl. Note that neither contains an abuse of notation, but both seamlessly integrate into the geometric algebra calculus. Moreover, it is possible to exten the vector erivative to general multivectors an take the erivative F(x) of any multivector-value function F(x) with respect to a vector argument. As a vector the position of in any geometric algebra expression is not arbitrary, since the geometric prouct is not commutative. As a consequence it no longer suffices to etermine the target of the operator. The following rule of notation may help to isambiguate things. In the absence of brackets, acts on the quantity to its immeiate right. When is followe by brackets, it acts on all of the terms insie the brackets. When acts on a multivector, which is not on its right, the scope will be signifie by overots. If has a subscript, then that inicates the variable with respect to which the erivative has to be taken. Consier the erivative of a neste function. Let u(v(x)) be such a function, with u, v an x vectors. Then, by the chain rule x u = u x i ei = u v j v j x i ei = x ( v(x) v )ú(v(x)). (4) With that it is possible to efine the irectional erivative of any multivector fiel in the irection of a vector a in terms of the geometric prouct as (a )F(x) = 1 2 ( a F(x) + aḟ(x) ). (5) 2.2 Directe Integration Consier a curve C in n-imensional space. Furthermore, assume that C has the parametric representation x(s) with enpoints a = x(α) an b = x(β). Even though there exist multiple alternative efinitions, for the purpose of this work let us efine the irecte line integral in terms of the scalar integral by

4 4 Carsten Cibura an Dietmar Hilenbran The term C β F(x)x = α F(x) x s. (6) s x = x s = e(x)s (7) s is calle irecte line element. The argument x of e often is suppresse. Three things are notable about this efinition. First, the vector e(x) is the tangent vector of the curve C in point x, spanning the tangent space there. Seconly, the measure x preserves a sense of irection, since it is efine as a vector-value infinitesimal quantity. An finally, the prouct between the integran an the measure is to be rea as a geometric prouct. That is, for example, for any vector fiel u(x) the integral can be split into C u(x)x = u(x) x + u(x) x. (8) C C In analogy to (7) one etermines the irecte line elements for coorinate curves in higher imensions by x k = x s k sk = e k s k. (9) From these the irecte area element of a surface S = {x(s 1,s 2 )} can be foun to be 2 x = x 1 x 2 = e 1 e 2 s 1 s 2, (10) an the irecte surface integral efine in terms of an iterate scalar integral by S β1 β2 F 2 x = Fx 1 x 2 = s 1 (s 1 ) s 2 Fe 1 e 2. (11) S α 1 α 2 (s 1 ) In n imensions the grae n irecte hyper-volume element n x is enote by X an the grae (n 1) hyper-surface element n 1 x by S. Because on a close curve C, irecte line elements occur pairwise with opposite signs, it is intuitively clear that x = 0. (12) C One of the most powerful applications of this is a generalization of Stokes theorem, which relates ifferentiation an integration on manifols. Theorem 1 (Generalize Stokes Theorem). Let V be a vector manifol with bounary V, L(A) a multivector-value function of a multivector argument A. Then V L( X) = L(S). (13) V We omit the proof here an refer the reaer to [2] or [5]. Let us point out, though, that the orientation of the manifol as well as its bounary is taken care of by the efinition of the irecte measures X an S, respectively.

5 Geometric Algebra Approach to Flui Dynamics 5 One important consequence of Stokes theorem is the fact that the path integral along a close curve in any graient fiel is zero. In orer to see this, consier the function L(A) = f A with f an A vectors in a two-imensional manifol V an f the graient of some scalar-value function w, i.e. f = w = w. Then f x = ḟ ( 2 x) V V = (ḟ ) 2 x V = ( w) 2 x = 0. (14) V Here, 2 x has grae two, serving as a pseuoscalar for the tangent space of V. By basic rewriting rules available in geometric algebra, it can be use to swap the inner an outer prouct, which we i in line two, above. 3 Flui Dynamics Consier a flui as a mass of particles moving in (n-imensional) space. As a particle is move aroun by external forces an by bouncing off other particles, its position x(x 0,t) = (x 1 (x0 1,...,xn 0,t),...,xn (x0 1,...,xn 0,t)) at a certain time t epens on its initial position x 0 = (x0 1,...,xn 0 ). An equivalent escription of the flui flow is to represent it as a vector fiel as it exists at a given time t. u(x,t) = x(x 0,t) t (15) = (u 1 (x,t),...,u n (x,t)), (16) where we obtain (16) from (15) by solving for x 0. However, the particle escription has its use an we will let ϕ t (x) = ϕ(x,t) enote the position of a particle initially at position x after time t. This implies ϕ(x,0) = x. ϕ is calle the flui flow map. As a physical entity the flow not only obeys the rules that apply to vector fiels in general, but also has to yiel to a number of physical principles. The change of any component of u with time implicitly epens on all the other components, as well as explicitly on time. Thus t u(x,t) = (u ) u + u t, (17) as can be verifie by applying the chain rule. We write D Dt = (u )+ t, an operator which is calle the material erivative. It takes into acount that the flui is moving an the particle position changes with time.

6 6 Carsten Cibura an Dietmar Hilenbran The law of conservation of mass hols, stating that ρ t + (ρu) = 0, (18) where ρ = ρ(x,t), the scalar-value mass ensity of the flui. There is a position- an time-epenent scalar-value function p(x, t), pressure, which escribes a force acting on any arbitrary surface insie the flui, in the irection of the surface normal. While pressure is a force acting locally, most real worl fluis are subject to global forces such as gravity, acting on the whole volume of flui. These forces are calle boy forces an enote b. The above properties hol for all flui flows, but there is a number of restrictions that can be place on a flow to create interesting an important special cases. An ieal flui is a flui in which no internal friction occurs. Therefore it oes not have viscosity. Note that in nature ieal fluis o not exist. However, they permit a simplification of certain concepts that can be extene later on. In an ieal flui, the law of balance of momentum hols, stating that ρ Du Dt = p + ρb. (19) A flui flow is calle incompressible, if its mass ensity is constant following the flow, i.e. Dρ Dt = 0. Equivalently, u = 0 (see [2] for proof). A flui flow is calle homogeneous, if its mass ensity is constant in space. A flui flow is calle isentropic, if there is a scalar-value function w(x, t), calle enthalpy, with w = 1 p. (20) ρ Figuratively speaking, w is a measure of the heat content of a flui. It consists of the internal energy an pressure work. Internal energy is mae up by molecular rotation an oscillation, potential energy of chemical bons etc. an roughly proportional to the flui s temperature. Pressure work is the work that is one to establish the flui s volume agains pressure s influence trying to compress the flui. 4 Application of Geometric Algebra to Flui Dynamics Definition 1 (Circulation). Given a close contour C at time t 0 = 0, enote with C t = ϕ(c,t) the contour transporte by the flow at time t. Then the circulation aroun C t is Γ Ct = u x. (21) C t

7 Geometric Algebra Approach to Flui Dynamics 7 Theorem 2 (Circulation Theorem). In an isentropic flow not subject to any boy forces, Γ Ct is constant in time. Proof. Suppose x(s),0 s 1 is a parametrization of C. Then ϕ(x(s),t),0 s 1 is a paramtetrization of C t. Recalling the form of the irecte line element we write u x = t C t t u x s s = 0 t u(ϕ t(x(s)),t) s ϕ t(x(s))s 1 + u(ϕ t (x(s)),t) 0 t s ϕ t(x(s))s 1 Du(ϕ t (x(s),t)) = x 0 Dt 1 + u(ϕ t (x),t) 0 s u(ϕ t(x),t)s. (22) But the final term is equal to s [ u 2 (ϕ t (x),t) ] s, because vectors commute. On the other han C s f (s)s = C f = 0, if C is close. So the final term vanishes an we arrive at Du u x = x. (23) t C t C t Dt Finally, for isentropic flows ρ 1 Du p = w. Balance of momentum states that ρ Dt = p + ρb. Excluing boy forces (b = 0), one conclues Du Dt = w an t Γ C t = w x = 0, (24) C t because the path integral along a close curve in a graient fiel is zero, as was shown in (14). Definition 2 (Vortex Sheets, Lines an Tubes). A vortex sheet (resp. vortex line) is a surface S (resp. a curve C), which in each of its points is orthogonal to the vorticity plane ξ u of a flui flow u. A vortex tube consists of a surface, nowhere locally a vortex sheet, with vortex lines rawn through each point of its bouning curve an extening as far as possible in each irection. Theorem 3 (Helmholtz Theorems). In an isentropic flow without boy forces, i). if a surface (a curve) moving with the flow is a vortex sheet (a vortex line) at time t = 0, then it remains so for all time. ii). if C 1 an C 2 are two close curves encircling a common vortex tube, then u x = C 1 u x. C 2 (25)

8 8 Carsten Cibura an Dietmar Hilenbran This common value is calle the strength of the vortex tube. iii). the strength of a vortex tube is constant in time as the tube moves with the flow. Proof. Let S(x) be a vortex sheet parametrize by s 1 an s 2. Then S = S s 1 s 1 S s 2 s 2 = e 1 (x) e 2 (x)s 1 s 2 is the irecte area element of S at x. The orthogonality property between S an ξ can be written as ( u(x)) (e 1 (x) e 2 (x)) = 0, for all x. Integrating over a vortex sheet an taking the erivative with respect to time yiels ( u) S = u x = 0, (26) t S t S accoring to (13), (21) an (24), which proves i) for vortex sheets. The claim for vortex lines is proven by the fact that every vortex line is the intersection of two vortex sheets, locally. Consier a vortex tube. Let S 1 an S 2 be two arbitrary cross sections of the tube with bouning curves C 1 an C 2, respectively. Let S enote the surface between C 1 an C 2, i.e. the tube s sie. Then S 1, S 2 an S enclose a region V an by (13) 0 = ( ( u)) X V = ξ S S 1 S 2 S = ξ S + ξ S + ξ S S 1 S 2 S = ξ S + ξ S S 1 S 2 = u x + u x, (27) C 1 C 2 because S is a vortex sheet. An since S 1 an S 2 are oriente oppositely, with surface normals pointing outwar, this proves ii). iii) follows from the Circulation Theorem. After having shown that geometric algebra can simplify the representation an proof of a number of flui flow properties, we now present the stream function ψ in mulitple imensions as a grae function, with the grae varying with the imension of the respective flow. This is possible because of the fact that geometric algebra is a grae algebra an that the grae of an algebraic object can be seen as a variable on a scale, rather than as an immutable, constituting property. In two imensions, this function is uniquely etermine up to an aitive constant. The behaviour of the stream function on streamlines allows for an ajustment of this aitive constant, which, in turn, etermines the stream function an through it the two-imensional flow. The lack of such a simple etermination is one of the main problems in escribing a three-imensional flow. Fining an exploring constituting properties of the multivector-value stream function ψ in higher imensions woul greatly facilitate the unerstaning of general flui flows.

9 Geometric Algebra Approach to Flui Dynamics 9 Employing geometric algebra methos it is easy to efine a stream function for n-imensional, incompressible flows. Assume such a flow with velocity fiel u, containe in some region D that is simply connecte. There exists a function ψ(x,t) of (pure) grae n 2, which fulfills ψ = ui 1 n, (28) where I n enotes the unit pseuoscalar in n imensions. This can be seen by 0 = ( ψ) = (ui 1 n ) = ( u)i 1 n, (29) which is zero if an only if ( u) is zero, i.e. the flow is incompressible. Note that (28) oes not suffice to etermine ψ. For example, aing the curl of any grae (n 3) function g(x) to ψ, such that ψ = ψ + g, oes not change u. u = ( ψ )I n = ( ψ + g)i n But, if we fix the ivergence of ψ, i.e. ψ = m, then = ( ψ)i n = u. (30) ψ = (ψ + g) = m + ( g) m. (31) Specifying the ivergence of ψ is calle choosing a gauge. Together with (28), choosing a gauge etermines ψ. Note that in n = 2 imensions, ψ is a scalar quantity an therefore ψ is zero. A relationship between vorticity ξ an the stream function ψ can be establishe as follows. ξ = u = (( ψ)i n ) = [ (( ψ)i n )]I n I 1 n = [ (( ψ)i n I n )]I 1 n = ( 1) 1 2 n(n 1) ( ( ψ))i 1 n. (32) Of special interest is the behaviour of ψ on streamlines. Definition 3 (Streamlines). At a fixe time, a streamline is an integral curve of the velocity fiel u(x,t) of a given flui flow. If x(s) is a streamline parametrize by s, then x(s) satisfies

10 10 Carsten Cibura an Dietmar Hilenbran x = u(x(s),t), t fixe. (33) s Let x(s) be a streamline parametrize by s. Then by (4) s ψ(x(s),t) = sψ(x(s),t) = s (ẋ(s) x ) ψ(x(s),t) ( = u(x(s),t) ) ψ(x(s),t) = (u ) ψ. (34) 5 Conclusion In the present work we have shown that the highly complex fiel of flui ynamics is accessible by the methos of geometric algebra, which facilitates the unerstaning of many concepts by literally aing new imensions of consieration. Also, we have propose a efinition for a stream function in arbitrary-imensional flui flows as a grae function. It can be seen to be ientical to the known stream function in the special case of two-imensional flows. We showe how this stream function relates to the geometric algebra formulation of the flow fiel s curl, i.e. the flui flow s vorticity. References 1. Baylis, W. E.: Electroynamics A Moern Geometric Approach. Birkhäuser Boston (2004) 2. Cibura, C.: Derivation of Flui Dynamics Basics Using Geometric Algebra. Diplomarbeit, Technische Universität Darmstat (2007) 3. Sommer, G.: Geometric Computing with Cliffor Algebras Theoretical Founations an Applications in Computer Vision an Robotics. Springer-Verlag (2001) 4. Sprößig, W.: Flui Flow Equations with Variable Viscosity in Quaternionic Setting. In: Avances in Applie Cliffor Algebras, volume 17. Birkhäuser (2007). 5. Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambrige University Press, Cambrige (1984) 6. Gürlebeck, K., Sprößig, W.: Quaternionic Analysis an Elliptic Bounary Value Problems. Birkhäuser (1990) 7. Hestenes, D., Sobczyk, G.: Cliffor Algebra to Geometric Calculus. Kluwer, Dorrecht (1984) 8. Hilenbran, D., Zamora, J., Bayro-Corrochano, E.: Inverse Kinematics Computation in Computer Graphics an Robotics Using Conformal Geometric Algebra. (2005) 9. Scheuermann, G., Hagen,H.: Cliffor Algebra an Flows. In: T. Lyche an L. L. Schumaker (es.), Mathematical Methos for Curves an Surfaces, pp Vanerbilt University Press (2001) 10. Wareham, R., Cameron, J., Lasenby, J.: Applications of Conformal Geometric Algebra in Computer Vision an Graphics In: Computer Algebra an Geometric Algebra with Applications, volume 3519 of Lecture Notes in Computer Science. Springer Verlag (2004)