. This paper investigates the onset of turbulence in incompressible viscous uid ow over a at plate by looking at the pressure gradients implied by the Blasius solution for laminar uid ow and adjusting the predicted ow, leading to a mathematically predictable ow separation in the boundary layer and the onset of turbulence (including both transition and fully turbulent regions - both with and without the presence of a at plate). It then considers the implications for potential analytic solutions to the Navier-Stokes Equations of the fact that it is possible to predict turbulence and a singularity for many ows (at any velocity). 1. AMS Subject Classi cation 35Q30, 76D05, 76D10 2. Introduction This paper investigates the onset of turbulence in incompressible viscous uid ow over a at plate by looking at the pressure gradients implied by the Blasius solution for laminar uid ow and adjusting the predicted ow, leading to a mathematically predictable onset of turbulence. Section 3 looks at the equations governing viscous incompressible ow in general and more speci cally for steady state ow in a boundary layer over a at plate. It then looks at the general characteristics of boundary layer ow over a at plate, introduces the Blasius solution modelling this ow and notes the observed pressure drop for ow over a cylinder. Section 4 looks at the Blasius solution in more detail, investigates the horizontal pressure gra- dients implied by the Blasius solution and adjusts the Blasius ow accordingly, leading to a predictable zero ow point close to the plate, resulting in ow separation and the onset of tur- bulence. This section also shows the applicability for the Blasius solution in free ow, without a physical plate boundary. Section 5 looks at the conclusions to be drawn - including both for predicting turbulence in geometries where Blasius can be directly applied but also more generally in free ow regions where we can say it is possible to predict the onset of turbulence (and a singularity) as a direct consequence of ow separation for many ows (at any initial velocity) - eliminating the possi- bility of analytical solutions to the Navier-Stokes equations in a number of situations. 3. Preliminary - Incompressible Flow Equations and Incompressible Flow Boundary Layer Observations 3.1. Incompressible Flow Equations The equations governing incompressible homogeneous newtonian uid ow in all of space RNN =2,3, are [1, p. 2]: Dv Dt=p+rv(Navier-Stokes for  >0 or Euler for = 0). divv= 0 (x;t)2RN[0;1) (Incompressibility). v t=0=v0; x2RN(Initial Conditions). wherev(x;t) is the uid velocity, p(x;t) is the scalar pressure,Dv Dtis the convective derivative (ie the derivative along the particle trajectories); D Dt= t+PN j=1vj xj The gradient operator ris: r= x1;:::; xNt; and the Laplace operator  is:  =PN j=12 x2 j = is the kinematic viscosity. ( is the viscosity, the density). For 2-dimensional steady state ow (ie no variation with time), these equations reduce to: (usingxforx1,yforx2,uforx1 t,vforx2 t) xNavier-Stokes:  uu x+vu y =p x+2u x2+2u y2 (1) yNavier-Stokes:  uv x+vv y =p y+2v x2+2v y2 (2) Incompressibility:u x+v y= 0 (3) For most practical purposes, a scale analysis of these equations eliminates a number of terms from the above equations, resulting in a boundary layer that has no pressure variation in the y direction and a pressure variation in the xdirection impressed from the external ow. In this paper, we are looking at the (small) pressure variations that do exist (and the implications) and so below we will look at a di erent approach. 3.2. Incompressible Flow over a Flat Plate Firstly observing the characteristics of incompressible uid ow over a at plate (steady state) as shown in gure 10-81 from [2, p. 514](not reproduced here): The key points of interest are a continually increasing boundary layer thickness (boundary layer edge being de ned as the point at which u = 0.99U) withu yalways the same sign, as well as the onset of turbulence as shown by the transition and turbulent regions (with corresponding increases on boundary layer thickness). If we now refer to the paper of Blasius (NACA translation) [3, p. 3], noting that Blasius uses for the boundary layer thickness, then we see that there will be a small pressure gradient across the boundary layer ( in the y direction - of the order of the square of the boundary layer thickness for a steady state ow). For most practical purposes this pressure gradient (resulting in a small pressure di erential across the boundary layer) is ignored - however we will not ignore it for the purposes of this paper. More importantly, we shall investigate the pressure pro le implied by the Blasius solution along the boundary layer (in the x direction). 3.3. Incompressible Flow over a cylinder In his paper, Blasius also applied this analysis to a ow over a cylinder. In this case, there is an additional term (related to the curvature of the cylinder) which generates a larger pressure gradient. An indication of the pressure di erential in laminar ow over a cylinder is given in the widely available graphs showing the di erence between theoretical and actual pressure coecients measured along the surface of a cylinder in a moving uid - a good example is in gure 10- 64 from [2, p. 504] (not reproduced here). The key points to notice here are the clearly increasing pressure di erential between theoreti- cal pressure coecient and laminar ow experimental results (up to boundary layer separation. Note that the pressure is measured on the surface of the cylinder) consistent with an increasing pressure gradient in the boundary layer as well as the much smaller pressure di erential seen in the turbulent boundary layer results. 4. Mechanism for the Onset of Turbulence 4.1. Blasius Solution Blasius in his paper (see[3]) provides an approximation for laminar ow over an in nite at plate by ignoring the small pressure variations developed along and across the boundary layer (approximating that any pressure pro le is impressed on the boundary layer by the external ow). This approximation works well for many applications, but does not predict the onset of turbulence. The reduced set of boundary layer equations that Blasius used were: (u t+uu x+vu y) =(U t+UU x) +2u y2 u x+v y= 0 Where U is the x component of the external ow. For the steady state situation this reduces to: (uu x+vu y) =(UU x) +2u y2 u x+v y= 0 Blasius used these equations to approximate a numerical solution (based on a similarity variable approach -=y(U x)1 2) for the ow velocities in a boundary layer. This solution has been shown to be usefully accurate experimentally. See gure 10-99 from [2, p. 523](not reproduced here). We can now investigate the implied x direction pressure gradient in a boundary layer by taking the Blasius results for velocity in the xdirection and (numerically) calculating the pressure pro- le by using the full Navier-Stokes equations (i.e. equations 1, 2 and 3 at the start of the paper). By using the Blasius values from [4] and matlab, based on an in nite plate in water owing at 1ms1, over a grid of 100,000 points in the x direction (up to x=0.6 so near observed turbulence) and 1000 points in the y direction (up to y = 0.004 or the edge of the boundary layer at x=0.5) - enough points to give us a good indication of the characteristics of the pressure gradient, if not enough to give us reliably accurate values - we arrive at the following results (see gure 1 below): Figure 1. Calculated Pressure Gradient and Horizontal Velocity The key points to note from this gure relevant to this paper are the continuously positive pressure gradients in the x direction (i.e continuous adverse pressure gradients -p x>0 for all values of y for large enough values of x) and the continually decreasing u value as x increases. We can also look in more detail at the pressure pro le in the region where <2:5 - i.e closer to the plate and away from the edge of the boundary layer (in the more 'linear' part of the Blasius graph above) - (see gure 2 below): This detail shows more clearly the adverse pressure gradients and velocities close to the plate, away from the boundary layer edge. It is important to note that these are calculated pressure gradients and velocities from the Blasius approximation (which assumed no pressure gradients generated inside the boundary layer). The presence of an adverse pressure gradient in boundary layer ow is a necessary but not suf- cient condition for ow separation. We can visualise the pressure gradient here as a kind of virtual di user. Figure 2. Calculated Pressure Gradient and Horizontal Velocity Detail <2:5 4.2. Adjusting the Blasius Solution with the Implied Pressure Gradients. It is now instructive to adjust the standard Blasius Solution by adding the calculated pres- sure gradient and nding the implied adjusted ow velocities. Considering the full (x-direction) steady-state Navier Stokes equation:  uu x+vu y =p x+2u x2+2u y2 We can rearrange this expression to evaluate the adjusted velocity gure when we superimpose the implied pressure gradient onto the Blasius solution. It is important to note that the expres- sion will be used to evaluate the change in variables. Rearranging: u x= p x+ 2u x2+2u y2 (u)vu y(4) If we now look at the magnitudes of the quantities involved (note these are all close to the plate - the magnitudes are signi cantly di erent away from the plate): Water:103103 u x: 105 p x: 106 2u x2: 105 2u y2: 107Note: Small due to the adverse the adverse pressure gradient. v: 1011 u y: 102 With those magnitudes in mind, a reliable rst approximation for the adjustedu xvalue (hence adjusteduvalue, by numerical integration along the xdirection) can be calculated numerically using the following reduced expression: u x= p x (u) A small adjustment to this expression (including an additional factor to allow for some in uence from the discarded terms) is: u x= p x (u1:2) The results of this calculation (using the same grid as before. Matlab code available on re- quest.) are shown in gure 3 below: The key observation from these results is that the adjusted velocity reduces to zero - rst for ow adjacent to the plate with the zero ow point moving further from the plate as xincreases. If we consider the concepts of marginal and massive separation (see [5, p. 403-411] for details) and use thexvalue as the S parameter, then we can see that the the rst zero ow point is a limit of incipient separation. It marks the start of a marginal separation (separation and reattachment) region (the transitional ow region) with a change to massive separation as S becomes large enough (the fully turbulent ow region - marked with a signi cant increase in the boundary layer thickness and a reduced pressue drop). In addition, we can see that the above observations are consistent with stability theory (see [5, p. 415] for details), if we note that perturbations are created by the separations and reattachments in the marginal separation region (even if there no external perturbations). Stability Theory provides a useful description of transitional and turbulent ow - not necessarily an explanation. The above observations show us that turbulence close to a at plate is a consequence of ow separation (marginal and massive), due to the (small) pressure gradients implied by the Blasius Solution. Below we show the Blasius Solution approach extended to a free- ow example. The inevitability of this result can be seen in the nature of the adjustment expression above - Figure 3. Pressure Gradient and Adjusted Horizontal Velocity the appearance of uin the denominator means that as ubecomes small,u xincreases rapidly, leading to a sudden reduction of uto zero (as can bee seen in the gure). In short, the Blasius solution adjusted with the e ects of the implied pressure gradient leads to a guaranteed (and calculable) onset of turbulence. In addition, we can see below that this result can be extended to free owing incompressible uid as well. The reduction of horizontal uid velocity to zero leads to a mathematical singularity (as can be seen with uin the denominator in equation (4)), to be expected with the onset of turbulence. 4.3. Showing the Applicability of the Blasius Solution to Free Flowing Fluid (No Plates or Other Obstructions). It is useful to show the adjusted Blasius solution above can be applied in the more general case of viscous incompressible uid owing without obstructions - two uid regions owing with di erent horizontal velocities (di erent values of u) with initially zero vvelocities. If we refer to gure 4 below showing key parameter values in the case of free- owing incom- pressible uid with the initial ow values mentioned above, with the ows assumed to start at t= 0 andx= 0: The key points to note in the diagram are the signs ofu xandv yat points a and b in the diagram (close to the boundary between the two layers). The immediate conclusion is that both partial derivatives are zero at the boundary. Figure 4. Blasius Free Flow Applicability. In the case of u, this means that the value of udoes not vary at the boundary, and if we accept that for large x, the value of uwill approachu1+u2 2, then we can say that uat the boundary will be constant atu1+u2 2. In the case of v, it is useful to look at the arrows in the gure for an impression of the sense of the paths of the particle ows. In the lower region (lower velocity, uincreasing as we approach the boundary from below), then the value of vis positive, decreasing to zero as we approach the boundary. In the upper region (higher velocity, udecreasing as we approach the boundary from above), the value of vis also positive, increasing from zero at the boundary. This means that we can treat the boundary as a virtual at plate (zero thickness) with boundary layers above and below and use the Blasius and adjusted Blasius approaches detailed above for modelling the ows near the plate. The nal point to match with the earlier analyses is to set the zero uvalue atu1+u2 2, so that the the upper ow initial velocity isu1u2 2and the lower ow initial velocity is u1u2 2. The immediate observation from the above is that there will (eventually) be turbulence for all incompressible viscid ows starting with di erent initial velocities. 5. Conclusions The Blasius solution for incompressible uid ow over a at plate is a very useful tool for practi- cal applications, but by eliminating pressure it eliminates the possibility of predicting the onset of turbulence . By adding back the e ects of the pressure gradient, it is possible to predict mathematically the onset of turbulence as a feature of ow separation - consistent with existing approaches including marginal and massive separation and stability theory (although more work will need to be done to establish suitable accuracy). Due to the wide applicability of the Blasius solution (for many geometries, including for ow without obstructions or plates and for ows of any velocity), this result suggests that, due to the predicted onset of turbulence and associated singularity, there may not be analytic solutions to the Navier-Stokes equations in most cases.