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Venturi: Potential Flow G.Rozza and A.T.Patera in collaboration with D.B.P Huynh and NC Nguyen March 28, 2008 1 Engineering Motivation Flows in ducts, channels, and pipelines are of great interest in ﬂuid mechanics applications especially when ﬂows can be studied in a parametrized geometrical conﬁguration. This Worked Problem considers a 2D potential ﬂow into a Venturi channel and is an example of a design of a parametrized ﬂuidic device as an element of a more complex modular ﬂuidic system (if we adopt a more complex ﬂuid model). In this Worked Problem we illustrate the calculation of pressure and velocity by the Bernoulli Theorem and the curvy geometry parametrization. We also illustrate some features dealing with potential ﬂows (i ) an error bound on velocity and pressure (and not only on the potential solution) and (ii ) visualization of the ﬂow ﬁeld (streamline) and pressure contours. 2 Physical Principles The physical principles illustrated in this Worked Problem deal with ideal ﬂuid mechanics (non-viscous ﬂuids) modelled by potential ﬂows and based on the Bernoulli Theorem. Velocity and pressure are inﬂuenced by the channel/constriction conﬁguration (i.e. height and length of the throat and radius of curvature of the connection). Gravitational eﬀects or other force ﬁelds could be applied. The Venturi tube is an example of the application of Bernoulli’s principle, in the case of incompressible ﬂow through a tube or pipe with a constriction in it. The ﬂuid velocity must increase through the constriction to satisfy the equation of continuity, while its pressure must decrease due to conservation of energy. The gain in kinetic energy is supplied by a drop in pressure or a pressure gradient force. The eﬀect is named after Giovanni Battista Venturi, (1746–1822), an Italian physicist. The limiting case of the Venturi eﬀect is choked ﬂow, in which a constriction in a pipe or channel limits the total ﬂow rate through the channel because the pressure cannot drop below 1 Γ3 Γ9 Γ4 Γ8 (4 − µ3 , 1) (4 + µ2 + µ3 , 1) Γ5 Γ7 (0, (0, 1) (8 + µ2 , 1) Γ2 Γ10 Γ1 R1 Γ11 (0, (0, 0) (8 + µ2 , 0) Γ14 Γ13 Γ12 (4, (4, 0) (4 + µ2 , 0) (4, 1 − µ3 ) (4, (4 + µ2 , 1 − µ3 ) Γ6 (4, µ1 + µ3 ) (4, (4 + µ2 , µ1 + µ3 ) (4 + µ3 , µ1 ) (4 + µ2 − µ3 , µ1 ) Figure 1: Parametrized Geometry Figure 2: Domain Boundaries zero in the constriction. Choked ﬂow is used to control the delivery rate of water and other ﬂuids through valves. Examples of the Venturi eﬀect are everywhere: in the capillaries of the human circulatory system; in large cities where wind is forced between buildings; in inspirators that mix air and ﬂammable gas in barbecues, gas stoves, Bunsen burners and airbrushes; in water aspirators that produce a partial vacuum using the kinetic energy from the faucet water pressure; in steam siphon using the kinetic energy from the steam pressure to create a partial vacuum; in atomizers that disperse perfume or spray paint (i.e., from a spray gun); in ﬁreﬁghting nozzles and extinguishers using foam; in compressed air operated industrial vacuum cleaners; in injectors (or sometimes called ejectors). Some carburetors are built use the Venturi eﬀect to suck gasoline into an engine’s air intake, and some vaporizers are designed to use the Venturi eﬀect to optimize eﬃciency. 3 Problem Description We consider the physical domain Ω(µ) shown in Figure 1. Here x = (x1 , x2 ) denotes a point in Ω(µ), non-dimensionalized with respect to height of the inlet L. Note that a tilde˜denotes dimensional quantities, and the absence of a tilde signals a non-dimensional quantity. We identify in Figure 2 the region R1 , representing the Venturi channel (inlet, connection, throat, connection, oulet). In this problem the boundary segments Γ3 , Γ5 , Γ7 , Γ9 are curved (all other boundary seg- ments and internal interfaces are straight lines). The segment Γ3 is given by the parametriza- tion x1 4 − µ3 1 0 µ3 0 cos (πt) = + , x2 1 − µ3 0 1 0 µ3 sin (πt) O1 (µ) Q1 (µ) S 1 (µ) where t ∈ [0, 1/2]. The segment Γ5 is given by the parametrization x1 4 + µ3 1 0 µ3 0 cos (πt) = + , x2 µ1 + µ3 0 1 0 µ3 sin (πt) O2 (µ) Q2 (µ) S 2 (µ) 2 where t ∈ [1, 3/2]. The segment Γ7 is given by the parametrization x1 4 + µ2 − µ3 1 0 µ3 0 cos (πt) = + , x2 µ1 + µ3 0 1 0 µ3 sin (πt) O3 (µ) Q3 (µ) S 3 (µ) where t ∈ [3/2, 2]. The segment Γ9 is given by the parametrization x1 4 + µ2 + µ3 1 0 µ3 0 cos (πt) = + , x2 1 − µ3 0 1 0 µ3 sin (πt) O4 (µ) Q4 (µ) S 4 (µ) where t ∈ [1/2, 1]. The Venturi element could be integrated into a more complex modular system. We consider here P = 3 parameters. Here µ1 , µ2 , µ3 are geometry parameters deﬁned in Figure 1; µ1 is the height of the throat (i.e., the central narrow part of the Venturi channel), µ2 is the length of the narrow part of the channel, and µ3 is the radius used to smooth the connections between the inlet (and the outlet) with the central throat. The parameter domain is given by D = [0.25, 0.5] × [2, 4] × [0.1, 0.2]. φ−LU The non-dimensional potential is denoted ϕ(µ) and is deﬁned as ϕ(µ) = ee , where φ ee LU is the dimensional potential, LU is the reference potential and U is a dimensional reference ˜ velocity. We will use the inlet velocity vin as reference in the following. The governing equation for ϕ(µ) is a generalized Laplacian: ∂ 1 0 ∂ − ϕ(µ) =0 in R1 , ∂xi 0 1 ∂xj κ1 ij with summation (i, j = 1, 2) over repeated indices. Velocity is given by the gradient of the potential function (being a conservative ﬁeld): ˜ ˜ ˜ ˜ v = vin ϕ(µ), where vin is the inlet dimensional velocity. By ﬁxing the pressure pin into ˜ a point (i.e. for example pin = 1 at the inlet) and having computed the velocity ﬁeld in ˜ ˜ all the domain v = vin ϕ(µ) (computing the gradient of the potential function) we may ˜ 1˜ v 1 get the pressure p by applying Bernoulli Theorem: p + 2 ρ(˜in )2 | ϕ(µ)|2 = pin + 2 ρ(˜in )2 , ˜ ˜ v ˜ where ρ is the density (also the Bernoulli equation can be non-dimensionalized). The non- dimensional velocity is given by v = ϕ(µ) and the non-dimensional pressure is given by p = (˜ − pin )/( 1 ρvin ) = (1 − | ϕ(µ)|2 ). p ˜ 2 ˜˜ 2 We must also impose interface and boundary conditions. On all internal interfaces (in- terior boundaries of regions), we impose continuity of the potential ϕ(µ) and its gradient ∂ ni κij ϕ(µ), where ni and ei denote normal and tangential unit vectors. ∂xj We show in Figure 2 the boundaries of the domain. On boundary Γ11 we impose homoge- neous Dirichlet conditions ϕ(µ) = 0, while on boundaries Γ2 , Γ3 , Γ4 , Γ5 , Γ6 , Γ7 , Γ8 , Γ9 , Γ10 , 3 Figure 3: Finite Element Mesh ∂ Γ12 , Γ13 and Γ14 we impose homogeneous Neumann conditions, ni κij ϕ(µ) = 0 (i.e., zero ∂xj velocity). In addition we impose non-homogeneous Neumann condition, ∂ϕ ni κ1 ij (µ) = 1 on Γ1 , ∂xj (i.e., imposition of the velocity at the inlet). For this problem the output of interest is provided by the visualization of velocity ﬁeld (by streamlines) and/or pressure contour ﬁeld. The error bound is computed on the pressure and on the velocity. This problem is then modeled by the P1 ﬁnite element (FE) discretization over the trian- gulation shown in Figure 3; the FE space contains Nt = 3137 degrees of freedom. This FE approximation is typically too slow for many applications, and we hence approximate the FE prediction for the output and ﬁeld variable by the reduced basis (RB) method. The user can obtain the RB prediction for the output and ﬁeld variable (visualization) — as well as a rigorous error bound for the diﬀerence between the RB and FE predictions — through our webserver. (Users who wish to run on their own computers and who have already downloaded our rbMIT software package can also create the RB approximation on their own computer from the rbU ﬁle.) 4 Additional Information for Visualization and Velocity/Pres- sure Pointwise Calculation 4.1 Vis_RB_Bernoulli and Online_RB_pointwise_Bernoulli The special visualization function provided with this problem Vis_RB_Bernoulli (i ) plots not only the potential solution but also the velocity ﬁeld by streamlines and the pressure contours and (ii ) computes an error estimator on the velocity and on the pressure and not only on the potential solution. The general call for the function is Vis_RB_Bernoulli(probname,mu) where probname='venturi' and mu is the vector parameter [mu1,mu2,mu3]. In addition, 4 [u, v, p, DeltaNV, DeltaNP]=Online_RB_pointwise_Bernoulli(probname, mu, xo) is the function to call to get pointwise quantities into an internal point (xo=[x,y]) or more points (xo=[x1,y1;x2,y2;x3,y3,...]) of the domain. Just specify the value (mu) of the parameter and 'probname'. As regards outputs: u is the (pointwise) horizontal velocity; v is the (pointwise) vertical velocity; and p is the (pointwise) pressure at the selected point(s). DeltaNV and DeltaNP are the L1 associated pointwise error bounds on velocity and pressure, respectively. The functions Vis_RB_Bernoulli and Online_RB_pointwise_Bernoulli are provided in the rbMIT_data folder along with the other data and functions related to this Worked Problem. 4.2 Some Provided Pre-Computed Solutions We provide in [u, v, p, DeltaNV, DeltaNP, varargout{1}]=Online_RB_pointwise_Bernoulli(... ... probname, mu, 'preset',varargin{1},varargin{2}) a set of “pre-computed” solutions on crucial points in the domain to extract velocity and/or pressure proﬁles. Two options are available: (i ) varargin{1}=0 and varargin{2}=x2; (ii ) varargin{1}=x1 and varargin{2}=0. With varargin{1}=0 and varargin{2}=x2 some pre-computed solutions are provided at an imaginary line at a constant height in the throat. The value of the height can be, for exam- ple, x2 = 0.05, 0.1, 0.15, 0.20, ..., 0.45. With varargin{1}=x1 and varargin{2}=0 some pre-computed solutions are provided on an imaginary line representing a section of the throat. The value of x2, for example, can be 4.0, 4.25, 4.5, . . . , 7.75, 8.0, and it should be contained inside the domain consistently with the value chosen for mu2; varargout{1}=xo. 5 Pedagogy Questions Q1. Predict/conﬁrm the shape of the velocity proﬁle and the pressure ﬁeld using a 1D model based on continuity equation and Bernoulli’s theorem. Hint: You can consider 3 diﬀerent sections — the inlet, the middle of the throat, and the outlet — and then compute the pressure drop and the pressure recovery. To answer this question you will need to use the additional function Online_RB_pointwise_Bernoulli provided to compute pointwise velocity/pressure and the associated L1 error bound.) Q2. What is the pressure behavior and the velocity proﬁle at high curvature points in the throat-inlet connection and in the throat-outlet connection? Hint: To answer this question you will need to use the additional function 5 Online_RB_pointwise_Bernoulli provided to compute pointwise velocity/pressure and the associated L1 error bound.) Q3. How does the height of the throat (i.e., mu3 parameter) aﬀect the low pressure at the “apex” of the curve in the transition region? Q4. How long does it take the ﬂow to become 1D in the throat region? What is the inﬂuence (if any) of the length of the throat (i.e., mu2 parameter)? 6