1 Similarity Analysis

Transcription

1 ME43A/538A/538B Axisymmetic Tubulent Jet 9 Novembe 28 Similaity Analysis. Intoduction Conside the sketch of an axisymmetic, tubulent jet in Figue. Assume that measuements of the downsteam aveage axial velocity pofile ae made downsteam of the jet exit at vaious locations which satisfy x/d > to 2, whee D is the jet diamete. In paticula, the velocity maxima ū m and the jet half-width l ae obtained fom the measuements. Fo this axisymmetic jet, the maximum velocity ū is found along the jet axis, =. The half-width of the jet at a paticula location in x is defined as the distance in y whee the aveage jet velocity has dopped to one-half of the peak velocity; i.e., it is defined mathematically at the downsteam location x as (efe to Figue ): u m (x) = =l(x) 2. x 2 x l 2 x l u m /2 u(x,) u m u 2m u 2m /2 u(x 2,) Figue : Sketch of an axisymmetic jet with measuements of the pofiles of the aveage axial velocity ū at the downsteam locations x and x 2. Now assume that measuement of the mean axial velocity pofile ae made, in paticula, at locations x and x 2, i.e., ū(x, ) and ū(x 2, ) ae measued. The peak velocities, u m (x ) = u m and u m (x 2 ) = u 2m, and the half-widths, l(x ) = l and l(x 2 ) = l 2, ae detemined. The mean velocities ū nomalized by u m ae then plotted on the same figue vesus nomalized by l, as seen in Figue 2. Plotted in this manne, the data collapse onto the same cuve, i.e., ū(x, ) u m = F(/l ) = ū(x 2, ) u 2m = F(/l 2 ), o, moe geneally, = F(/l) u m fo any downsteam position x which is geate than D to 2D. A flow which has this popety is called self-simila o self-peseving. An impotant featue of such a flow is that, once U m (x), l(x), and F ae found, then is known. We will find that it is easie to detemine these quantities, eithe fom expeiments o fom theoy and numeical simulations, than it is to find

2 itself. Futhemoe, knowing u m (x) and l(x) will allow us to make seveal infeences about the flows. Note that wakes, mixing layes, and bounday layes also become self-simila. Futhemoe, the concept of self-similaity is also useful in heat tansfe and a numbe of othe technical fields..2 Self-similaity applied to the time-aveaged equations u/u m x x 2 /l Figue 2: Sketch of an axisymmetic jet with measuements of the pofiles of the aveage axial velocity ū at the downsteam locations x and x 2. As has been established in class, the bounday laye equations hold fo the case of an axisymmetic, tubulent jet when x/d > to 2, depending on the chaacteistics of the nozzle. These equations ae: ū + v x = (u v ) () x + ( v) =. (2) The similaity assumptions fo the two components of the aveage velocity and fo the Reynolds stess ae: ( ) U(x) = F (3) l(x) ( ) v(x, ) U(x) = G l(x) Hee U(x) and l(x) ae defined by: u v (x, ) U 2 (x) ( ) = H. l(x) U(x) = = ū(x, ) max U(x) = =l 2, i.e., 2

3 fo a given downsteam distance x, U(x) is the maximum velocity (which is along the centeline), and l is the adial distance at which ū equals /2 of its maximum value (l is often efeed to as the half-width). When these assumed foms ae plugged into the vaious tems in the momentum and continuity equations, the following deivatives esult: x = U F U l l ηf, whee ( ) denotes the deivative of a function with espect to its agument, and η = /l. Also = U l F (u v ) = U 2 l ( v) = U l η η η (ηh) η (ηg). Finally, plugging these expessions into the oiginal equations gives, afte multiplying the momentum equation by l/u 2 and the continuity equation by l/u: Next assume powe law foms fo U and l, i.e., ( U ) l F 2 l ηff + F G = (ηh) (4) U η η ( U ) l F l ηf + (ηg) =. (5) U η η U(x) = Ax n l(x) = Bx m, whee it is expected that m, n >. Note that the coefficients in Equations (4) and (5) must be constant fo the equations to be consistent; i.e., since the last tems ae only functions of η, then each tem in the equations can only be a function of η and not x. This implies that so that m = and U l U = nax n Ax n Bx m = nbx m = constant, l(x) = Bx. (6) Note that the coefficients of the fist two tems in Equations (4) and (5) ae nb. It has also been established in class that the momentum flux acoss any vetical plane pependicula to the flow diection is constant, i.e., M = 2πρ ū 2 (x, )d = 2πρl 2 U 2 ηf 2 dη = constant. Note that the last integal (ove η) in this equation is constant, since it is an integal ove a given function fo fixed limits. Theefoe, l 2 U 2 = B 2 x 2 A 2 x 2n = constant, 3

4 so n =, and U(x) = Ax. (7) With these constant coefficients, and the values found fo n and m, Equations (4) and (5) educe to: BF 2 BηFF + F G = (ηh) (8) η η BF BηF + η (ηg) =. (9) η Note that we still have two equations with thee unknowns, the unknowns now being F, G, and H. We still have to deal with the tubulence closue poblem and eliminate one of the unknowns befoe solving the equations..3 Self-similaity applied to the mass flux Befoe consideing tubulence modeling fo this poblem it is useful to conside the implications of Equations (6) and (7) on the mass flux m(x). Using a definition simila to that fo momentum flux, the mass flux though the vetical plane pependicula to the flow diection is: m(x) = vetical plane 2π ρ(v n) da = ρ = 2πρ d dθ }{{} da d = 2πρ U(x)l 2 (x) ηf(η) dη,() whee the similaity fom fo, Equation (3), has been used. ηf(η) dη must be a constant, then, using Equations (6) and (7), Since the definite integal m(x) U(x)l 2 (x) x x 2 = x, that is, the mass is the jet inceases popotional to x. This is a diect esult of entainment of fluid into the jet, is an essential featue of jets, and can be impotant in poblems such as nonpemixed combustion. This has implications egading entainment. As the flow goes fom x to x 2, the mass flux in the jet inceases by an amount popotional to x 2 x due to entainment (see Figue 3). 2 Tubulence modeling The mixing length assumption fo the Reynolds stess is: with the tubulent viscosity ν T given by u v = ν T ν T = c UL. Hee c is a constant, to be detemined using theoy, numeical simulations, o laboatoy data; U is a chaacteistic velocity and L a chaacteistic length scale of the tubulence. Using the similaity 4

5 (fuel) entainement (fuel) mass flux (mixtue) inceased mass flux x x 2 Figue 3: Sketch of the entainment pocess in a jet flowing fom the downsteam location x to x 2 as its mass inceases by an amount popotional to x 2 x. assumption fo the Reynolds stess, and identifying the velocity scale U as U and the length scale L as l, then u v = U 2 H = c Ul 2 F (UF) = c U η, o H = c F η. With this in the momentum equation, Equation (8), it becomes, finally: BF 2 BηFF + F G = c ( η F ). () η η η The exact solution to these equations have been found to be: U(x) = F(η) = ( + 8 Aη2 ) 2 with η = l(x) whee U(x) v(x, ) U(x) = and dl η = G(η) = dx 2 U(x) = =l(x) 2 ] [ 8 Aη2 [ + 8 Aη2 ] 2 u v U 2 (x) = H(η) = 2 A c η ] 3 [ + 8 Aη2 In ode fo data, then U(x) = =l(x) ( + 8 A)2 = 2, then A. = 3.3. Futhemoe, to match laboatoy c =.256, and 5

6 l(x) = Bx with B =.848 U(x) = Ax with M A = 7.4 ρ. Figue 4 gives plots of /U, v(x, )/U, and u v /U as functions of η = /l. A ough idea of the laboatoy data is also given in the figue. Note that the ageement with the data is faily good, patly since the constants have been adjusted to optimize ageement fo this flow. The ageement of the model with the data begins to beak down at aound η > 3, whee the data points dop below the model pediction, and whee also the tubulent flow has become vey intemittent. This intemittency is not accounted fo in the modeling. Some consistency checks can be made egading the modeling assumptions. Fo example, x U x = x (Ax ) = x (Ax ) = U x. So the diffeential scale fo in x is just x itself, i.e., l x = x. The diffeential scale in was found to be l = Bx. Theefoe, l = Bx l x x = B =.848, which is athe small and oughly consistent with the assumption that l /l x. Futhemoe, with some calculus and algeba, it can be found that v U.27. This is consistent with the implied assumption that v m /ū m. Finally, some futhe calculus and algeba leads to u v m U 2 =.7, consistent with the conclusion that u v m/u 2. 3 Summay Axisymmetic Jet Results simila to those fo the axisymmetic jet can be obtained fo plane jets, axisymmetic and plane wakes, axisymmetic and plane mixing layes, and axisymmetic and plane buoyant plumes. What follows is a summay of the main esults fo the axisymmetic, tubulent jet. Pincipal assumptions: the flow in thin, i.e., ( ) x ( ) high Reynolds numbe: Re = Ul ν the esult is the bounday laye equations: x + ( v) = + v x y = (u v ) 6

7 Figue 4: Plots of /U(x), v(x, )/U(x), and u v /U(x) as functions of η = /l(x). With the momentum flux M(x) defined as M(x) = 2πρ ū 2 (x, )d, then, fo x/d > to 2, M(x) is constant in x. The momentum flux is one of the pincipal featues defining a jet. On the othe hand, with the mass flux m(x) defined as m(x) = 2πρ d, then, fo x/d > to 2, m(x) inceases popotional to x. This is closely elated to the entainment pocess. With a closue assumption fo the Reynolds stess ρu v in tems of a tubulent viscosity, solutions can be obtained fo ū, v, and u v giving faily good ageement with the data. Howeve, some adjustable constants have to popely chosen to obtain the ageement. This is indicative of the tubulence modeling found in the vaious commecial codes: to some degee they amount to sophisticated cuve-fitting. 7