Chapter 1 - Basic Equations

Transcription

1 Marine Hydrodynamics, Sring 2005 Lecture Marine Hydrodynamics Lecture 2 Chater 1 - Basic Equations 1.1 Descrition of a Flow To define a flow we use either the Lagrangian descrition or the Eulerian descrition. Lagrangian descrition: Picture a fluid flow where each fluid article caries its own roerties such as density, momentum, etc. As the article advances its roerties may change in time. The rocedure of describing the entire flow by recording the detailed histories of each fluid article is the Lagrangian descrition. A neutrally buoyant robe is an examle of a Lagrangian measuring device. The article roerties density, velocity, ressure,... can be mathematically reresented as follows: ρ (t), v (t), (t),... The Lagrangian descrition is simle to understand: conservation of mass and Newton s laws aly directly to each fluid article. However, it is comutationally exensive to kee track of the trajectories of all the fluid articles in a flow and therefore the Lagrangian descrition is used only in some numerical simulations. (t) υ r Lagrangian descrition; snashot 1

2 Eulerian descrition: Rather than following each fluid article we can record the evolution of the flow roerties at every oint in sace as time varies. This is the Eulerian descrition. It is a field descrition. A robe fixed in sace is an examle of an Eulerian measuring device. This means that the flow roerties at a secified location deend on the location and on time. For examle, the density, velocity, ressure,... can be mathematically reresented as follows: v( x, t),( x, t),ρ( x, t),... The aforementioned locations are described in coordinate systems. In we use the cartesian, cylindrical and sherical coordinate systems. The Eulerian descrition is harder to understand: how do we aly the conservation laws? However, it turns out that it is mathematically simler to aly. For this reason, in Fluid Mechanics we use mainly the Eulerian descrition. y r ( x, t) r r υ ( x, t) x Eulerian descrition; Cartesian grid 2

3 1.2 Flow visualization - Flow lines Streamline: A line everywhere tangent to the fluid velocity v at a given instant (flow snashot). It is a strictly Eulerian concet. Streakline: Instantaneous locus of all fluid articles that have assed a given oint (snashot of certain fluid articles). Pathline: The trajectory of a given article P in time. The hotograh analogy would be a long time exosure of a marked article. It is a strictly Lagrangian concet. Can you tell whether any of the following figures ( [1] Van Dyke, An Album of Fluid Motion 1982 (.52, 100)) show streamlines/streaklines/athlines? 3

4 1.3 Some Quantities of Interest Einstein Notation Range convention: Whenever a subscrit aears only once in a term, the subscrit takes all ossible values. E.g. in 3D sace: x i (i = 1, 2, 3) x 1, x 2, x 3 Summation convention: Whenever a subscrit aears twice in the same term the reeated index is summed over the index arameter sace. E.g. in 3D sace: a i b i = a 1 b 1 + a 2 b 2 + a 3 b 3 (i = 1, 2, 3) Non reeated subscrits remain fixed during the summation. E.g. in 3D sace a i = x ij nˆj denotes three equations, one for each i = 1, 2, 3 and j is the dummy index. Note 1: To avoid confusion between fixed and reeated indices or different reeated indices, etc, no index can be reeated more than twice. Note 2: Number of free indices shows how many quantities are reresented by a single term. Note 3: If the equation looks like this: (u i )(ˆx i ), the indices are not summed. Comma convention: A subscrit comma followed by an index indicates artial differentiation with resect to each coordinate. Summation and range conventions aly to indices following a comma as well. E.g. in 3D sace: Scalars, Vectors and Tensors u i u 1 u 2 u 3 u i,i = = + + x i x 1 x 2 x 3 Scalars magnitude density ρ ( x, t) ressure ( x, t) Vectors (a i x i ) magnitude direction velocity v ( x, t) /momentum mass flux Tensors (a ij ) magnitude direction orientation momentum flux stress τ ij ( x, t) 4

5 1.4 Concet and Consequences of Continuous Flow For a fluid flow to be continuous, we require that the velocity v( x, t) be a finite and continuous function of x and t. v i.e. v and are finite but not necessarily continuous. Since v and v <, there is no infinite acceleration i.e. no infinite forces, which is hysically consistent Consequences of Continuous Flow Material volume remains material. No segment of fluid can be joined or broken aart. Material surface remains material. The interface between two material volumes always exists. Material line remains material. The interface of two material surfaces always exists. Material surface fluid a fluid b Material neighbors remain neighbors. To rove this mathematically, we must rove that, given two articles, the distance between them at time t is small, and the distance between them at time t +δt is still small. 5

6 v Assumtions At time t, assume a continuous flow ( v, << ) with fluid velocity v( x, t). Two arbitrary articles are located at x and x + δ x(t), resectively. Result If δ x(t) δ x 0 then δ x(t + T ) 0, for all subsequent times t + T. v r Proof v x + {υ (x x v )δt} v δ x v δx (t + δt) v v x +δ x v x + r δ x v + v v r ) + δ x υ (x {(υ (x v ))δt} After a small time δt: The article initially located at x will have travelled a distance v( x)δt and at time t + δt will be located at x + { v( x)δt}. The article initially located at x+δ x will have travelled a distance ( v( x)+ δ x v( x)) δt. (Show this using Taylor Series Exansion about( x, t)). Therefore after a small time δt this article will be located at x + δ x + {( v( x)+ δ x v( x))δt}. The difference in osition δ x(t + δt) between the two articles after a small time δt will be: δ x(t + δt) = x + δ x + {( v( x)+ δ x v( x))δt} ( x + { v( x)δt}) δ x(t + δt) = δ x +(δ x v( x))δt δ x Therefore δ x(t + δt) δ x because v is finite (from continuous flow assumtion). Thus, if δ x 0, then δ x(t + δt) 0. In fact, for any subsequent time t + T : t+t δ x(t + T ) δ x + δ x vdt δ x, t and δ x(t+t ) 0as δ x 0. In other words the articles will never be an infinite distance aart. Thus, if the flow is continuous two articles that are neighbors will always remain neighbors. 6

7 1.5 Material/Substantial/Total Time Derivative: D/Dt A material derivative is the time derivative rate of change of a roerty following a fluid article. The material derivative is a Lagrangian concet. By exressing the material derivative in terms of Eulerian quantities we will be able to aly the conservation laws in the Eulerian reference frame. Consider an Eulerian quantity f( x, t). The time rate of change of f as exerienced by a article travelling with velocity v is the substantial derivative of f and is given by: Df( x (t),t) f( x + v δt, t + δt) f( x,t) r = lim (1) f ( x,t) Dt δt 0 δt article r x (t) f (x r + r υ δ t, t + δt) r r x ( t ) + υ δt Performing a Taylor Series Exansion about ( x,t) and taking into account that v δt = δ x, we obtain: f( x, t) f(x + v δt, t + δt) = f( x, t)+ δt + δ x f( x, t)+ O(δ 2 )(higher order terms)(2) From Eq.(1, 2) we see that the substantial derivative of f as exerienced by a article travelling with v is given by: The generalized notation: Df f = + v f Dt D + v }{{} Dt } {{} Lagrangian Eulerian 7

8 Examle 1: Material derivative of a fluid roerty G ( x, t) as exerienced by a fluid article. Let denote a fluid article. A fluid article is always travelling with the local fluid velocity v (t) = v( x,t). The material derivative of a fluid roerty G ( x, t) as exerienced by this fluid article is given by: D G G = + v G Dt }{{ }}{{}}{{} Convective Lagragian Eulerian rate of change rate of change rate of change Examle 2: Material derivative of the fluid velocity v( x, t) as exerienced by a fluid article. This is the Lagrangian acceleration of a article and is the acceleration that aears in Newton s laws. It is therefore evident that its Eulerian reresentation will be used in the Eulerian reference frame. Let denote a fluid article. A fluid article is always travelling with the local fluid D v( x,t) velocity v (t) = v( x,t). The Lagrangian acceleration Dt as exerienced by this fluid article is given by: D v v = + v v Dt }{{}}{{}}{{} Convective Lagragian Eulerian acceleration acceleration acceleration 8

9 1.6 Difference Between Lagrangian Time Derivative and Eulerian Time Derivative Examle 1: Consider an Eulerian quantity, temerature, in a room at oints A and B where the temerature is different at each oint. Point A: 10 o Point C: T t Point B: 1 o At a fixed in sace oint C, the temerature rate of change is T which is an Eulerian time derivative. Examle 2: Consider the same examle as above: an Eulerian quantity, temerature, in a room at oints A and B where the temerature varies with time. DT T = + v υ fly T Point A: 10 o Dt t Point B: 1 o Following a fly from oint A to B, the Lagrangian time derivative would need to include DT T the temerature gradient as both time and osition changes: Dt = + v fly T 9

10 1.6.1 Concet of a Steady Flow ( 0) A steady flow is a strictly Eulerian concet. Assume a steady flow where the flow is observed from a fixed osition. This is like watching D from a river bank, i.e. = 0. Be careful not to confuse this with Dt which is more like following a twig in the water. Note that D = 0 does not mean steady since the flow could Dt seed u at some oints and slow down at others. = 0 Dρ Concet of an Incomressible Flow ( Dt 0) An incomressible flow is a strictly Lagrangian concet. Assume a flow where the density of each fluid article is constant in time. Be careful not to confuse this with ρ = 0, which means that the density at a articular oint in the flow is constant and would allow articles to change density as they flow from oint to oint. Also, do not confuse this with ρ = const, which for examle does not allow a flow of two incomressible fluids. 10