New estimation of near exit streamwise instabilities on high speed liquid jets

Transcription

1 ILASS Americas 28th Annual Conference on Liquid Atomization and Spray Systems, Dearborn, MI, May 2016 New estimation of near exit streamwise instabilities on high speed liquid jets G.A. Jaramillo and S.H. Collicott School of Aeronautics and Astronautics Purdue University West Lafayette, IN USA Abstract Linear stability analysis (LSA) of liquid jets has traditionally been performed with hyperbolic secant or Gaussian velocity profiles. A different approach is presented here, in which a shear flow profile, near the jet exit (0.2 < x/d < 1.0) is used to analyze streamwise instabilities in high speed liquid jets at 150 < Re δ2 < 670. Here x is the streamwise location, D is the exit jet diameter, and Re δ2 is the Reynolds number based on the momentum thickness of the boundary layer. The Rayleigh equation is solved using a hyperbolic tangent velocity profile to perform a temporal and a spatial stability analysis (TSA and SSA respectively). Results show agreement with respect to experimental results, by Portillo et al. [1], of better than 15 % for the nearexit streamwise instabilities wavelength at the jet surface. The momentum thickness is the chosen reference length to scale estimated instabilities wavelength values. A discussion about the benefits of the Blasius equation versus CFD to determine the former is offered. Among some of the advantages of the hyperbolic tangent profile is that the integration of the respective ODE is made on a real path instead of the complex path which is typical of LSA solutions for the other profiles. Also, in this work is discussed the impact of the change of variables and a mathematical transformation known as the Riccati transformation to simplify numerically the derivation and the solution of the Rayleigh equation to perform respective LSA s. Corresponding Author: gjaramil@purdue.edu. He is also affliated to the School of Mechanical Engineering at Universidad del Valle in Colombia.

2 Introduction One traditional approach on estimating liquid jets instabilities is based on the Linear Stability Analysis (LSA) theory which involves the solution of the Rayleigh equation. In this approach a parallel, inviscid and incompressible flow is assumed and a velocity profile is analytically defined to model the jet physics. Previous works, see Portillo et al. in 2011 [1], Drazin and Reid in 2004 [2], Criminale et al. in 2003 [3], and Schmid and Henningson in 2001 [4], have chosen hyperbolic secant and Gaussian velocity profiles to model circular jet flows, while in coaxial jets, another velocity profile like the hyperbolic tangent is the chosen one, see Perrault-Joncas and Maslowe in 2008 [5] and Talamelli and Gavarini in 2006 [6]. However, in both cases the stability analysis is oriented far away from the exit of the jet nozzle and LSA fails to estimate streamwise liquid jet instabilities near it. See Portillo et al. [1] and Portillo in 2008 [7]. A different approach is made in this work, defining a hyperbolic tangent velocity profile near the nozzle exit, to estimate nondimensional streamwise circular liquid jet instabilities wavelengths (λ). The momentum thickness (δ2), has been chosen as the reference length to scale λ from LSA, see Portillo et al. [1] and Hoyt and Taylor in 1977 [8]. An illustration of relevant length scales for two different velocity profiles, is shown in Fig. 1, based on their streamwise location within the jet. The Blasius equation is a common reference to estimate apriori δ 2 and therefore, to scale λ. However, a discussion about its validity is shown here and a comparison based on Computational Fluid Dynamics (CFD) simulations is displayed as well. The hypothesis to support this approach is the strong dependence the momentum thickness exerts on streamwise liquid jets instabilities and, a hyperbolic tangent velocity profile is chosen to model the shear effect close to exit of the nozzle. Previously, hyperbolic tangent profiles have been used to model planar shear flows, as it is seen in Criminale et al. [3] and Michalke in 1964 [9]. Nondimensional wavelengths λ from LSA s are scaled with δ 2 estimations from the Blasius equation and CFD simulations then, compared and shown in the results and discussion section. Experimental results to make comparisons with, are taken from Portillo et al. [1] for 150 < Re δ2 < 670 in the near exit region at (0.2 < x/d < 1.0). Problem statement To solve the Rayleigh equation in LSA circular liquid jets, it is required to assume a velocity profile. While a Gaussian or hyperbolic secant velocity profile spans with the jet diameter, at its respective streamwise location (η ), a hyperbolic tangent velocity profile may span with boundary layer (δ ), displacement (δ 1), or momentum (δ 2) thicknesses, as it is shown in Figs. 1 and 2. In this work, δ 2 is the reference length to scale λ, as it seen in Fig. 2. Figure 2. Velocity profiles used to model circular jets in LSA, and scaled with respective reference lengths. Figure 1. Circular liquid velocity jet profiles at different streamwise locations. Velocity transition to specify yet. In Fig. 2 the horizontal axis is the dimensional jet velocity U as a function of the spanwise variable y, scaled with respect the dimensional jet centerline velocity U o. Vertical axes are the normalization of the y coordinate, on the left and blue; with respect a selfsimilar spanwise variable (η, or jet diameter at the respective streamwise location); And on the right and green, with respect to δ 2. Hyperbolic secant and Gaussian profiles are read with the left and blue axis, while the hyperbolic tangent velocity profile is read with the right and green axis. The Rayleigh equation is a second order Ordinary Differential Equation (ODE), which becomes 2

3 a complex ODE when it is intended to perform a LSA throughout the perturbation method. In the LSA context, a velocity profile is defined and that profile requires an inflection point to satisfy the Rayleigh s inflection-point theorem such as the Fj rtoft s theorem as well, to find an instability on the base flow, see Criminale et al. [3], Schmid and Henningson [4], and Riahi in 2000 [10]. The inflection point on the velocity profile becomes a singularity when the Rayleigh equation is integrated and a complex path must appear to avoid that singularity, see Schmid and Henningson[4]. However, Michalke [9, 11] showed that avoiding the complex path integration is possible depending on the assumed velocity profile. He worked with a hyperbolic tangent velocity profile to model planar shear flows, and in this work, the same velocity profile is defined to model the flow condition at the near exit of a high speed water jet. A 2D velocity profile is chosen, based on perturbations on the jet surface are several orders of magnitude smaller than the radius of the circular water jet. Once it is clear the procedure to solve the Rayleigh equation for a specific velocity profile, LSA s are performed solving the Rayleigh equation for different parameters of real wavenumbers (α), and complex phase speeds (c),(for Temporal Stability Analysis - TSA), or real perturbation frequencies (β),(for Spatial Stability Analysis - SSA), finding the case which shows the higher growth rate and highlighting the corresponding most unstable λ. In hydrodynamics stability theory, is common practice to scale λ with δ2. Previous works on high speed circular jets, see Portillo et al. [1], Hoyt and Taylor [8], and McCarthy and Molloy in 1974 [12], have used the Blasius equation to estimate apriori δ2 at the exit of jet nozzle. Authors in this work, have taken a different approach using CFD simulations to estimate δ2. Single phase axysimmetric CFD simulations of a laminar flow, are performed to estimate velocity profiles at the very exit of the jet nozzle with residuals criteria of continuity and momentum around 10 6, using second order upwind numerical schemes. A difference on δ2 estimated values, given by the Blasius equation, and CFD simulations, is shown and discussed in next sections. Hyperbolic tangent velocity profile The chosen velocity profile in this work is the hyperbolic tangent (1). From the mathematical standpoint, the singularity point, which is the inflection point, is located at the symmetry axis of the velocity profile, which allows to solve the Rayleigh equation without integrating it through the complex path. Instead, a real path integration is done up to the singularity point without including it. An advantage is that a mathematical transformation known as the Riccati transformation can be used, and an increase in the accuracy is achieved from the numerical perspective. ( ) 1 U (y) = {1 + tanh (y)} (1) 2 where, U(y) is the nondimensional axial velocity of the jet, as a function of the nondimensional spanwise variable y. Riccati transformation and change of variable The main advantage with the Riccati transformation is the order decreasing of the ODE therefore, a reduction in the mathematical complexity to integrate the Rayleigh equation. Additionally, a change of variable is done using s = tanh(y), and an increase on the numerical accuracy is reached due to the exactness at the boundaries, see Michalke [9]. Instead of dealing with numerical values close to the exact value at the boundary, this change of variable allows to use an exact value on the boundary. To see it, compare boundary conditions (BC) in next sections for derived set of equations with and without the Riccati transformation. The Riccati transformation is shown next, G = 1 dφ φ dy (2) where, φ is a complex stream function, and G is the new complex function to find, after solving the respective ODE. Through the Riccati transformation, the Rayleigh equation for a TSA is rewritten next and it becomes a first order ODE, dg dy + G α2 ( d 2 U dy 2 U c ) φ = 0 (3) where, α is the streamwise real wavenumber, and c is the complex phase speed. In the same way for a SSA, the Rayleigh equation due to the Riccati transformation is shown next, dg dy + G γ2 ( d 2 U dy 2 U β γ ) φ = 0 (4) where, γ is the streamwise complex wavenumber, and β is the real perturbation frequency. Equations (3,4) are nondimensional. It is shown next, derived expressions for each stability analysis with and without the Riccati transformation, including the change of variable when the 3

4 Riccati transformation is implemented, and the complex Rayleigh equation becomes a system of two second order ODE without the Riccati transformation, or a system of two first order ODE if the transformation is applied. Temporal stability analysis The TSA is a mathematical tool which allows to seek the transition to a convectively unstable (CU) flow. This type of analysis will not define by itself if the flow is CU, but it will show if there is a possible transition to that status, see Fig. 5. In this work, TSA and SSA are compared to estimate streamwise instabilities wavelengths. The Rayleigh equation to perform a TSA is shown next, (U c) ( d 2 φ dy 2 α2 φ ) d2 U dy 2 φ = 0 (5) This Rayleigh equation for a TSA is derived with and without the Riccati transformation leading to a two ODE system with real and imaginary components. Notice next, the increase on mathematical complexity on the derivation without the Riccati transformation nor the proposed change of variable. Without the Riccati transformation, dφ r dy dφ i dy (y ± ) 0 (10) (y ± ) 0 (11) There is a singularity point at y = 0, which is the inflection point in the hyperbolic tangent profile presented in (1). Due to the symmetry of the hyperbolic tangent profile, as it is seen in Fig. 2, equations (6,7) are integrated from y + to y 0 + and from y to y 0. Then, the appropriate numerical solution is the one which shows the least difference from these two real paths of integration at y 0, when they are compared close to the singularity point. With the Riccati transformation, The derivation of real and imaginary components of the Rayleigh equation using the Riccati transformation and a change of variable is shown here, dg r ds = α2 G 2 r + G 2 i 1 s 2 2s2 s 2 + 4c 2 i (12) The real component of the Rayleigh equation is presented next, dg i ds = 2G rg i 1 s 2 4sc s 2 + 4c 2 i (13) d 2 φ r dy = α2 φ r + [ 1 tanh 2 (y) ] [ ] φ r 2 tanh 2 (y) + c i φ i tanh (y) tanh 2(y) 4 + c 2 i (6) where, φ r and φ i are the real and imaginary components respectively, of the stream function φ, and c i is the imaginary component of the complex phase speed, indicating the rate at which the wavelike perturbation group speed is changing. The imaginary component of the Rayleigh equation is shown below, d 2 φ i dy = α2 φ i + [ 1 tanh 2 (y) ] [ ] φ i 2 tanh2 (y) c i φ r tanh (y) with following BC, tanh 2(y) 4 + c 2 i (7) φ r (y ± ) 0 (8) φ i (y ± ) 0 (9) where, G r and G i are the real and imaginary components of the G function. This system is solved using two real integration paths from s 1 to s 0 + and another one from s 1 to s 0. At the singularity point s = 0, a numerical solution is not defined, and a similar procedure previously described, is implemented. The respective BC s of this system of first order ODE are shown next, G r (s ±1) = αs (14) G i (s ±1) = 0 (15) A MATLAB code is executed using a Runge- Kutta integration scheme to solve the Rayleigh equation with the Riccati transformation and the change of variable, and Fig. 3 shows a comparison between these authors code and Michalke [9] for a TSA. In the horizontal axis is specified the streamwise real wavenumber (α), and in the vertical axis is defined the growth rate, which in a TSA is given by αc i. Table 1, shows a numerical comparison at the highest growth rate between the code written by these authors and Michalke [9]. The higher difference was less than 0.1 %. 4

5 d 2 φ i dy = γ2 r φ i + 2γ r γ i φ r γ 2 i φ i + A [ H (γ i φ r + γ r φ i ) B ( γ 2 i φ i γ r γ i φ r )] H 2 + (Bγ i ) 2 (18) where, A = [ tanh(y)][1 tanh 2 (y)] (19) B = (1/2) (1 + tanh (y)) (20) H = Bγ r β (21) Figure 3. TSA validation. Growth rate vs. streamwise wavenumber for a hyperbolic tangent velocity profile. References αc i α Jaramillo and Collicott Michalke [9] Difference [%] Table 1. Comparison for a TSA of a hyperbolic tangent profile. Spatial stability analysis This mathematical tool allows to identify an absolute unstable flow (AU), decomposing the phase speed in a real perturbation frequency (β) and a complex wavenumber (γ). This analysis is more demanding than a TSA, from the computational standpoint. The Rayleigh equation required to perform a SSA is shown below, ( U β ) ( d 2 ) φ γ dy 2 γ2 φ d2 U dy 2 φ = 0 (16) The derived real and imaginary components of the Rayleigh equation without the Riccati transformation are presented next. Without the Riccati transformation, d 2 φ r dy = γ2 r φ r 2γ r γ i φ i γi 2 φ r + A [ H (γ r φ r γ i φ i ) B ( γ r γ i φ i + γi 2φ )] r H 2 + (Bγ i ) 2 (17) with γ r and γ i being the real and imaginary components of γ respectively, and the same BC s required for a TSA without the Ricatti transformation, see (8,9,10,11). With the Riccati transformation, The derived components of the Rayleigh equation (16), using the Riccati transformation and a change of variable are shown next, dg r ds = γ2 r γi 2 G2 r + G 2 i 1 s 2 2s ( ) [ ( ) ] γr 2 + γi 2 (1 s) γ 2 r + γi 2 2βγr [(1 s) (γr 2 + γi 2) 2βγ r] 2 + 4β 2 γi 2 (22) dg i ds = 2 (γ rγ i G r G i ) 1 s 2 4s ( ) γr 2 + γi 2 (βγi ) [(1 s) (γr 2 + γi 2) 2βγ r] 2 + 4β 2 γi 2 with following BC, (23) G r (s ±1) = γ r s (24) G i (s ±1) = γ i s (25) A results comparison is shown in Fig. 4, between the written code in MATLAB for this work, and Michalke [11] work, to perform a SSA of a hyperbolic tangent profile with the Riccati transformation, and a change of variable. The horizontal axis in Fig. 4, shows the real perturbation frequency (β) and the vertical axis spans the growth rate, given by γ i in a SSA. Table 2 highlights the highest difference between the code written in this work and the code by Michalke [11] is less than 1.2%. 5

6 Figure 5. LSA tools to estimate convective and absolute instabilities. Figure 4. SSA validation. Growth rate vs. frequency for a hyperbolic tangent velocity profile. References γ i β Jaramillo and Collicott Michalke [11] Difference [%] Table 2. Comparison for a TSA of a hyperbolic tangent profile. Convective and Absolute Instability Temporal and spatial stability analyses are mathematical tools that allow to identify the transition between convective and absolute instability (AU/CU) in a base flow, see Fig. 5. In 2008, Portillo [7] showed that an absolute pocket of instability near the exit of the jet nozzle causes the registered instabilities displayed in his work for different geometric aspect ratios of the nozzle. Even when LSA tools are meant to identify AU/CU, in this work TSA and SSA are also used to estimate first λ in high speed water jets. Momentum thickness The momentum thickness (δ 2) is a flow characteristic which allows to estimate other flow characteristics as drag, using the Von Karman expression for instance. But beyond that, δ 2 is not an actual length measurement in the flow, contrary to the boundary layer thickness δ, which it is. Furthermore, δ 2 is a reference length which compares a made up flow which has a constant and flat velocity profile, with the actual flow having a non-flat velocity profile throughout its entire cross section. The main characteristic about these two flows is they share the same momentum therefore, the cross sec- tion of the made up flow is smaller than the actual one, and in a circular jet flow, the reduction in the radius of the actual jet flow making the comparison with the invented flow, is δ2. One of the most common and widely spread estimations of δ2 is calculated for an incompressible laminar flow over a flat-plate given by Blasius, see Schlichting in 1979 [13], and also known as the Blasius equation, δ 2 = ν D U (26) where, ν is the kinematic viscosity of the fluid (in this case, is water), and D is the jet diameter at the exit of the nozzle. Previous works in circular jet flows have used the Blasius equation (26) to estimate δ2 at the exit of the jet nozzle, under the assumption that perturbations are small enough compared with the jet radius. Then, a 2D planar case can be supposed, and the flow at the exit of the circular jet nozzle may be analyzed as a flat-plate. In this work, single phase (water) CFD simulations will be used to estimate velocity profiles and δ2 at the exit of the jet nozzle. Finally, a comparison of δ2 estimations is shown and discussed in the results section of this work. Instability wavelength estimation method The new estimation of streamwise circular liquid jets instabilities near the exit of the nozzle is done through these steps: Deriving the set of equations to perform TSA and SSA. Performing TSA and SSA of a hyperbolic tangent velocity profile. 6

7 Estimating the momentum thickness based on computational and/or experimental tests. Scaling wavelength estimations given by each one of the performed stability analyses and make comparisons with experiments by Portillo et al. [1] The nondimensional instability wavelengths (λ) are estimated in different ways depending on the respective performed LSA. For a TSA: λ = 2π/α, and for a SSA: λ = 2π/γ r CFD simulations to estimate momentum thickness To estimate δ 2 at the exit of the jet nozzle, single phase laminar simulations of a 2D axysymmetric nozzle are performed. In Fig. 6 is shown a sketch of the region used to perform CFD simulations. Figure 6. Sketch of the 2D axisymmetric region used to perform CFD simulations. Drawing is not a scale. A finite volume method in ANSYS-FLUENT was implemented to run CFD simulations on two different nozzle geometries based on the nozzle aspect ratio and three different mesh grids, for each nozzle geometry, were used to verify numerical convergence. For L/D = 1, mesh grids have 4.4k, 8.8k and 19k grid points, and for L/D = 5, they have 10k, 21k and 44k grid points. Inlet velocities are estimated based on mass flow rates (ṁ ) and applied as BC at the inlet section of the geometry with water density (ρ ), and dynamic viscosity (µ ), at 20 o C. Results and discussion In this section, is shown λ estimations based on LSA and scaling with δ2 2 given by the Blasius equation (26), and CFD simulations. The comparison shown in Fig. 7 is based on experimental data taken by Portillo [7], for 14.6 < P < 115 and 0.10 < ṁ < 1.15, and in Fig. 8, for 70 < P < 85 and 0.30 < ṁ < 1.00, where P is the water pressure at the nozzle inlet, see Fig. 6. Pressure and ṁ units are psia and kg/s respectively. An uncertainty estimation on these calculations is included, and the introduction of a TSA modifying the hyperbolic tangent velocity profile as well. Uncertainty estimation The uncertainty estimation from experimental tests is specified by Portillo [7]. On the instabilities wavelength (λ ) measurement, the uncertainty value was 13.3 % for jet mean flow velocities up to 40 m/s, due to blurring in the image processing and the uncertainty magnitude for estimated Re δ2 is around 2%, where the momentum thickness δ2 is obtained by the Blasius equation (26), and jet velocity values are estimated based on ṁ, which uncertainty sources come from timing and quantity estimation of it. The uncertainty magnitude from LSA estimations of instabilities wavelengths ( λ num), is given by the comparison with respect to Michalke results, and it is specified around 1.2%, see Fig. 3 and 4 and tables 1 and 2. Next results show differences between momentum thickness estimation from the Blasius equation (26) and CFD simulations. That difference reaches values up to 45% in L/D = 1, and 57% in L/D = 5, see Fig. 7 and 8. That difference defines the momentum thickness uncertainty δ2, assuming the momentum thickness obtained from CFD simulations is the accurate one. However, δ2 = 11% including CFD results exclusively, and in the worst evaluated case, see Fig. 7 and 8. Next expression shows an estimation of the total uncertainty to calculate streamwise instability wavelengths. ± λ = ( λ num + δ 2) (27) where, λ 60% is the total uncertainty of the estimated λ if δ 2, estimated by the Blasius equation (26) is included, or λ 13% if only CFD results are involved to estimate δ 2. All variables in equation (27) are in percent. It is important to notice that the major source of error comes from the estimation on δ 2 which is δ 2. Additional work on this topic must be performed. Momentum thickness estimation Next figures show a comparison between CFD simulations and the Blasius equation (26) to estimate δ 2, for L/D = 1 in Fig. 7, and L/D = 5 in Fig. 8. In Fig. 7, is shown that the Blasius equation underestimates δ 2 in comparison with CFD results. A similar trend is shown if Fig. 8, where δ 2 is underestimated again by the Blasius equation (26) for a L/D = 5 nozzle. 7

8 Figure 7. Momentum thickness at different Re δ2 for a L/D = 1 nozzle. Although, in Fig. 7, CFD results show one unique trend, there are several points which do not collapse. This is explained due to the closeness of the nozzle exit to the upstream section change, within one diameter length, which numerically brings a source of instability due to a fast change on the pressure gradient and therefore, on the flow conditions. An improvement in the mesh grid distribution through the computational domain is demanded to reduce numerical errors. where Re δ2 has been recalculated for L/D = 1 and L/D = 5 respectively. It is noticed δ2 spans over a different range of Re δ2 depending on its estimation approach. Based on CFD results, new values of mean velocities are obtained and those are used to recalculate their respective Re δ2. In the L/D = 5 nozzle, as it is seen in Fig. 8, δ2 estimations through CFD simulations collapse better than in a L/D = 1 nozzle. In this case, the nozzle exit is located at five diameters from the upstream section change therefore, flow conditions are more stable from the numerical standpoint than for a L/D = 1 nozzle. While in the L/D = 5 nozzle, δ2 CFD estimations collapse better than in the L/D = 1 nozzle, the difference between δ2 estimated by Blasius and CFD simulations is greater than in the L/D = 1 nozzle, which could be interpreted as a rejection of the Blasius equation to estimate δ2 as the nozzle aspect ratio increases. It means that assumptions taken to apply the Blasius equation in the δ2 estimation are not longer valid. Modification of the hyperbolic tangent profile Before making a comparison with experimental λ, it is shown a TSA using modified hyperbolic tangent profiles. The reason to do this is to add a family of velocity profiles which could fit better to estimate λ. In Fig. 9, is displayed how the hyperbolic tangent velocity profile changes, based on equation (28) which modifies it. U = 1 2 {1 + tanh(fy)} = 1 (1 + s) (28) 2 where f is a factor greater than 1, modifying the hyperbolic tangent slope as it is shown in Fig. 9, and s = tanh fy The respective set of derived ODE using the Riccati transformation is presented next in equations (29) and (30), where the real and imaginary components are respectively shown. dg r ds = α2 G 2 r + G 2 i f (1 s 2 ) 2s2 f s 2 + 4c 2 i (29) Figure 8. Momentum thickness at different Re δ2 for a L/D = 5 nozzle. An appropriate comparison of δ2 between the Blasius equation (26), and CFD results, must include the respective and actual reference with Re δ2. One problem appears when δ2 has different values depending on the estimation method, because it will also change Re δ2 and that is shown in Fig. 7 and 8, dg i ds = 2G rg i f (1 s 2 ) 4sfc i s 2 + 4c 2 i (30) In Fig. 10, is shown a set of 6 different velocity profiles based on the hyperbolic tangent profile and modified as appears in equation (28) with 1 f 2. From this set of TSA, is chosen the one with f = 2 to find a λ. 8

9 The last one is used in the λ estimations for the L/D = 5 nozzle. Wavelength comparison In the L/D = 1 nozzle, a λ comparison for 110 < Re δ2blasius < 320 at 0.23 < x/d < 0.32 between experimental data and LSA estimations is shown next in Fig Also, a difference between wavelength estimations via scaling with δ 2 by the Blasius equation (26) and CFD simulations, is highlighted as well. Figure 9. Modified hyperbolic tangent profiles Figure 11. Wavelength comparison scaled with δ 2 estimated by Blasius equation (26), for a L/D = 1 nozzle and different LSA. See Fig. 12 for improved scaling Figure 10. Growth rate vs wavenumber for modified hyperbolic tangent profiles in a TSA The TSA for a hyperbolic tangent profile with 1 f 2 shows that f = 1 has the highest α and therefore, the smallest λ. On the opposite way, f = 2 has the smallest α thus, the highest λ while, for these set of profiles shown in Fig. 10, f = 1.2 has the highest growth rate. With the TSA for modified hyperbolic tangent profiles, is shown next, results from four different LSA to find λ and after scaling with δ2, estimating λ. These LSA are: TSA with f = 1 and λ = SSA with f = 1 and λ = TSA with f = 2 and λ = TSA with f = 1.2 and λ = In Fig. 11, the best λ estimation is done through a TSA with f = 2, with an error less than 22%. In all cases, λ estimated values decrease with the increase of Re δ2, and the trend is satisfactorily captured. Estimations of λ based on CFD simulations and their respective comparison with experimental data are shown in Fig. 12. The best λ estimation is given by a TSA with f = 2, where the highest error is less than 10%. In Fig. 12 is shown CFD results from a mesh grid with 8.8k grid points. In Fig. 13, a comparison is made between experimental and estimated λ by a TSA with f=2. The best fit on the λ estimation for the L/D = 1 nozzle, is reached with CFD results using a TSA with f = 2, having an error of 10% with λ = 13% and less than the experimental λ uncertainty. For the L/D = 5 nozzle,in Figs , is shown a λ comparison for 170 < Re δ2blasius < 300 at 0.23 < x/d < 0.75 between experimental data and LSA estimations. 9

10 Figure 12. Wavelength comparison scaled with δ 2 estimated by CFD, for a L/D = 1 nozzle and different LSA Figure 14. Wavelength comparison scaled with δ 2 estimated by Blasius equation (26), for a L/D = 5 nozzle and different LSA Figure 13. Best wavelength estimation for a L/D = 1 nozzle at 0.23 < x/d < 0.32 Experimental λ are sparse over the region and do not follow an unique trend and λ estimations using LSA do not match either with experimental data. In Fig. 14, it is shown the comparison of λ estimations, using different LSA with δ2 calculated by the Blasius equation (26) and the experimental λ measurements. In Fig. 14 the closest λ estimation is given by a TSA with f = 2, and an error margin around 57%. None of λ estimations captured the experimental data distribution. With a mesh grid of 21k grid points, Fig. 15 shows λ comparison between experimental and CFD results using different LSA. In Fig. 15, λ estimations, using a TSA with f = 1.2, have error margins around 37% without following the experimental data distribution neither. In Fig. 16 is shown that, the best fit for λ estimations is given by a TSA with f = 1.2 and a CFD approach to estimate δ 2. In any case, for a L/D = 5 nozzle, none LSA could capture the experimental λ distribution. Further experimental tests are required to verify such distribution of λ and additional comparison with the streamwise instabilities would be useful to improve the understanding about this method to estimate λ. In Fig. 17 is shown the comparison between experimental λ, nearest to the exit if the jet nozzle, and λ estimation using different LSA and CFD simulations to scale them with δ 2, at 150 < Re δ2 < 670. It is also seen, in Fig. 17, an agreement between λ estimation and experimental λ measurements for a L/D = 1 nozzle. Nevertheless, such agreement does not exist for a L/D = 5 nozzle at 390 < Re δ2 < 670. A SSA with a modified hyperbolic tangent profile is the next step to work on, which by definition, will show an absolute instability and therefore, will be more accurate than a TSA. However, with the procedure shown here to estimate λ is sound to rely on a TSA with modified hyperbolic tangent profiles as a first approach to reach that goal. Beyond becoming just another numerical exercise, the f factor models the shear effect at the exit of the nozzle and could be taken as a different way to tune and control δ 2 estimations given by the Blasius equation (26). Furthermore, going backwards with λ measurements near the exit of the jet nozzle, δ 2 estimations could be performed on the jet 10

11 Figure 15. Wavelength comparison scaled with δ 2 estimated by CFD, for a L/D = 5 nozzle and different LSA nozzle at the exit. Conclusions A good agreement between the new λ estimation method and experimental measurements is presented in this work with errors lower than the experimental uncertainty for a L/D = 1 nozzle. The momentum thickness based on the Blasius equation has to be reviewed for each aspect ratio of high speed water jet nozzles. The Blasius equation underestimates δ 2, and for both evaluated nozzles, λ estimations presented a lower error with δ 2 calculated through CFD simulations. For a L/D = 1 nozzle, a TSA with a modified hyperbolic tangent profile and a δ 2 estimation using CFD simulations, fits best experimental λ measurements with an error less than 10%. For a L/D = 5 nozzle, none LSA provided a suitable estimation to compare with experimental data. Additional work on experimental λ measurements is required, and streamwise instabilities comparison using this method as well, to verify its relevance. Acknowledgment Authors thankfully acknowledge the contribution with his experimental data to Ph.D. Enrique Portillo and the funding provided by Universidad del Valle, Fulbright and Colciencias. Figure 16. Best wavelength estimation for a L/D = 5 nozzle at 0.23 < x/d < 0.75 Nomenclature U velocity profile ν kinematic viscosity L nozzle length D diameter at the exit of the jet nozzle R radius at the exit of the jet nozzle δ boundary layer thickness δ 1 displacement thickness δ 2 momentum thickness λ streamwise instability wavelength Re δ2 Reynolds number based on δ 2 η spanwise selfsimilar jet diameter φ complex stream function G stream function after Riccati transf. α streamwise real wavenumber γ streamwise complex wavenumber β real perturbation frequency c complex phase speed δ 2 momentum thickness uncertainty λ total instability wavelength uncertainty inst. wavelength uncertainty from LSA λ num Subscripts r real component i imaginary component Blasius estimated by the Blasius equation (26) CFD estimated by CFD simulations Superscripts dimensional value 11

12 [12] M.J. McCarthy and N.A. Molloy. The Chemical Engineering Journal, 7:1 20, [13] H. Schlichting. Boundary-Layer Theory. Seventh edition, Figure 17. Wavelength estimation for both nozzles at 150 < Re δ2 < 670 References [1] J.E. Portillo, S.H. Collicott, and G.A. Blaisdell. Physics of Fluids, 23, [2] P.G. Drazin and W.H. Reid. Hydrodynamic Stability. Cambridge Mathematical Library, Cambridge. UK., second edition, [3] W.O. Criminale, T.L. Jackson, and R.D. Joslin. Theory and Computation of Hydrodnamic Stability. Cambridge University Press, Cambridge.UK, [4] P.J. Schmid and D.S. Henningson. Stability and Transition in Shear Flows. Advanced Mathematical Sciences. Vol 142. Springer, [5] D. Perrault-Joncas and S.A. Maslowe. Physics of Fluids, [6] A. Talamelli and I. Gavarini. Flow Turbulence Combustion, 76: , [7] J.E. Portillo. PhD thesis, School of Aeronautics and Astronautics. Purdue University, [8] B.J. Hoyt and J.J. Taylor. J. Fluid Mech., 83. part 1.: , [9] A. Michalke. J. Fluid Mech., 19: , [10] D.N. Riahi. Flow Instability. WIT Press, [11] A. Michalke. J. Fluid Mech.,, 23. part 3: ,