ENV5056 Numerical Modeling of Flow and Contaminant Transport in Rivers Reynolds Transport Theorem Asst. Prof. Dr. Orhan GÜNDÜZ
We are sometimes interested in what happens to a particular part of the fluid as it moves. Other times we may be interested in what effect the fluid has on a particular object or volume in space. Thus, we need to describe the laws governing fluid motion using both tem concepts (a given mass of fluid) and control volume concepts (a given volume). The analytical tool that helps us to shift from one representation to the other is called Reynolds Transport Theorem. Let B represent a fluid parameter (extensive property) and b represent the amount of that parameter per unit mass (intensive property): B = mb where m is the mass of the fluid. Thus, if the extensive property of the fluid is mass (B=m) then b=1. 2
B is proportional to the amount of mass considered whereas b is independent of mass. The amount of an extensive property that a tem contains, B is calculated as: B = ρbd Most laws of fluid motion involve a time rate of change of an extensive property of a fluid tem. So we have: db dt d = ρbd dt To formulate these laws in a control volume approach, we must have an expression for time rate of change of an extensive property within a control volume db dt d = ρbd dt Reynolds Transport Theorem provides the relationship between these two equations 3
Given below is a fixed control volume and a moving tem: a. b. The control volume is the stationary (fixed) volume within a pipe between sections 1 and 2 (green dash line). The tem is the fluid occupyingthe control volume at some initial time t. A short time later, t+δt, the tem has moved slightly to the right (black dash line). The fluid particles that coincided with section 2 of the control surface has moved a distance δl 2 =V 2 δt to the right. V 2 is the velocity of the fluid as it passes section 2. Similarly, fluid initially at section 1 has moved a distance δl 1 =V 1 δt to the right where V 1 is the velocity of the fluid as it passes section 1. 4
As shown in (b), the outflow from the control volume from time t to t+δt is denoted by volume II, the inflow as I and the control volume itself as CV. Thus, the tem at time t consists of the fluid in section CV. (SYS=CV at time t). At t+δt, the tem consists of the same fluid that now occupies sections CV-I and II (SYS=CV-I+II). The control volume remains as V at all times. If B is the extensive property of the tem, then the value of it for the control volume at time t is: ( ) = ( ) B t B t Since the tem and control volume coincide at this time. Its value at t+δt is: ( + δ ) = ( + δ ) ( + δ ) + ( + δ ) B t t B t t B t t B t t I II Now substractb (t) from both sides and divide by δt. 5
Thus, the change in the amount of B in the tem in the time interval δt divided by this time interval is given by: ( + ) ( ) ( + ) ( + ) + ( + ) ( ) δ B B t δt B t B t δt BI t δt BII t δt B t = = δt δt δt Since initially B (t) = B (t), we obtain: ( + ) ( ) ( + ) ( + ) δ B B t δt B t BI t δt BII t δt = + δt δt δt δt In the limit δt 0, the le hand side is equal to the me rate of change of B for the tem and is denoted by DB /Dt, the material derivative. We use the material derivative notation to denote that this time rate of change has a Lagrangianbehavior such that it represents time rate of change of the property with a given fluid particle as it moves along. 6
In the limit δt 0, the first term on the right hand side is equal to the me rate of change of B for the control volume and is denoted by B / t B ( t + δt) B ( t) B ρbd δt t t lim = = δt 0 The third term in the boxed equation represents the rate at which B flow out of the control volume across the control surface. The amount of B in region II is its amount per unit volume, ρb, times the volume δv=a 2 δl 2 =A 2 (V 2 δt). Thus, II ( ) ( )( ) B t + δt = ρ b δ = ρ b A V δt 2 2 II 2 2 2 2 where ρ 2 and b 2 are contantvalues of ρ and b across section 2. So the rate at which this property flows from the control volume is given by: i B out ( + δ ) B t t = lim II = ρ b A V δ t 0 δt 2 2 2 2 7
Similarly, the inflow of B into the control volume across section 1 during time interval δtcorresponds to that in region I and is given by the amount per unit volume times the volume δv=a 1 δl 1 =A 1 (V 1 δt). Thus, I ( ) ( )( ) B t + δt = ρ b δ = ρ b AV δt 1 1 I 1 1 1 1 where ρ 1 and b 1 are contantvalues of ρ and b across section 1. So the rate at which this property flows into the control volume is given by: i B in ( + δ ) B t t = lim I = δt 0 δt ρ b AV 1 1 1 1 Combining these equations gives: DB Dt B t i = + out B i B in 8
Or: DB Dt B t = + ρ2b2 A2V 2 ρ1b1 AV 1 1 This is a version of Reynolds Transport Theorem valid for some simple conditions such as a fixed control volume with one inlet and one outlet having uniform properties across the inlet and outlet with velocities normal to the sections 1 and 2. It is important to note that the time rate of change of B for the tem is therefore not necessarily the same as the time rate of change of B within the control volume since the inflow rate and outflow rate of the property of B for the control volume need not be the same. 9
In a more general sense, Reynolds Transport Theorem could be derived for a fixed arbitrary control volume with fluid flowing through it. The tem is again defined to be the fluid within the control volume at time t. A short time later, a portion of the fluid (Region II) exists the control volume and additional fluid (Region I) enters the control volume. 10
Although the logic is the same as before, the control volume can have more (or less) than one inlet and one outlet. The term B out can represent the net flowrateout of the control volume and could be evaluated by integrating (summing) of all contributions through each one of the infinitely small area elements of size δaon outlet portion of the control surface Cs out. In δt, the volume of fluid that passes across each area element is given by δv= δl n δawhere δl n = δlcosθis the height (normal to the base δa) of the small volume element and θ is the angle between the velocity vector and the outward pointing normal to the surface n 11
Since δl=v δt, the amount of property B carried across the area element δain the time interval δtis given by: ( cos ) δ B = bρδ = bρ V θδ t δ A The rate at which B is carried out of the control volume across the small area element δais denoted by δb out ( cos ) bρδ bρv θδt δ A δ Bout = lim = lim = b ρ V cos θδ A δ t 0 δt δ t 0 δt Integrating over the entire surface gives the total amount of property traveling out of the control volume. Bout = d Bout = bρv cosθ da cs out cs out 12
The quantity Vcosθis the component of velocity normal to the area element δa. From the definition of dot product this can be written as: V cos = V n So the above equation can be written as: Bout = b ρ V n da cs out ( ) Similarly, tjeinflow portion of the control surface Cs in can be used to derive the corresponding equation for B in Bin = bρv cosθ da = bρ V n da cs in cs in ( ) 13
The standartnotation states that the unit normal vector always points out of the control volume. Thus, the dot product of velocity and the unit vector has a negative sign. The value of cosθis therefore positive on outflowingportions of the control volume and negative on inflowing portions of the control volume. Therefore, the net flux of parameter B across the entire control surface is: csout csin cs ( ) ( ) ( ) Bout Bin = bρ V n da bρ V n da = bρ V n da 14
When we combine all terms in the general equation, we obtain: DB Dt B t ( ) = + b cs ρ V n da Or in a slightly different way by substituting: B = ρbd To give: DB Dt = ρbd + bρ ( V n) da t cs This is the general form of Reynolds Transport Theorem for a fixed, nondeforming control volume. 15
Reference Munson, B.R., Youn, D.F. And Okiishi, T.H. (1998). Fundamentals of Fluid Mechanics. Third Edition, John Wiley and Sons Inc., 877p. 16