An official website of the United States government
Abstract We study higher uniformity properties of the Möbius function$$\mu $$, the von Mangoldt function$$\Lambda $$, and the divisor functions$$d_k$$on short intervals$$(X,X+H]$$with$$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$$for a fixed constant$$0 \leq \theta < 1$$and any$$\varepsilon>0$$. More precisely, letting$$\Lambda ^\sharp $$and$$d_k^\sharp $$be suitable approximants of$$\Lambda $$and$$d_k$$and$$\mu ^\sharp = 0$$, we show for instance that, for any nilsequence$$F(g(n)\Gamma )$$, we have$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when$$\theta = 5/8$$and$$f \in \{\Lambda , \mu , d_k\}$$or$$\theta = 1/3$$and$$f = d_2$$. As a consequence, we show that the short interval Gowers norms$$\|f-f^\sharp \|_{U^s(X,X+H]}$$are also asymptotically small for any fixedsfor these choices of$$f,\theta $$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$$L^2$$. Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type$$II$$sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$$I_2$$sums.
more »
« less
Energy flux and high-order statistics of hydrodynamic turbulence
We use the Dyson–Wyld diagrammatic technique to analyse the infinite series for the correlation functions of the velocity in hydrodynamic turbulence. We demonstrate the fundamental role played by the triple correlator of the velocity in determining the entire statistics of the hydrodynamic turbulence. All higher-order correlation functions are expressed through the triple correlator. This is shown through the suggested triangular re-summation of the infinite diagrammatic series for multi-point correlation functions. The triangular re-summation is the next logical step after the Dyson–Wyld line re-summation for the Green's function and the double correlator. In particular, it allows us to explain why the inverse cascade of the two-dimensional hydrodynamic turbulence is close to Gaussian. Since the triple correlator dictates the flux of energy$$\varepsilon$$through the scales, we support the Kolmogorov-1941 idea that$$\varepsilon$$is one of the main characteristics of hydrodynamic turbulence.
more »
« less
- Award ID(s):
- 2009418
- PAR ID:
- 10499933
- Publisher / Repository:
- JFM
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Edition / Version:
- 1
- Volume:
- 975
- ISSN:
- 0022-1120
- Subject(s) / Keyword(s):
- turbulence theory, Navier–Stokes equations
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
A lubrication analysis is presented for the resistances between permeable spherical particles in near contact,$$h_0/a\ll 1$$, where$$h_0$$is the minimum separation between the particles, and$$a=a_1 a_2/(a_1+a_2)$$is the reduced radius. Darcy's law is used to describe the flow inside the permeable particles and no-slip boundary conditions are applied at the particle surfaces. The weak permeability regime$$K=k/a^{2} \ll 1$$is considered, where$$k=\frac {1}{2}(k_1+k_2)$$is the mean permeability. Particle permeability enters the lubrication resistances through two functions of$$q=K^{-2/5}h_0/a$$, one describing axisymmetric motions, the other transverse. These functions are obtained by solving an integral equation for the pressure in the near-contact region. The set of resistance functions thus obtained provide the complete set of near-contact resistance functions for permeable spheres and match asymptotically to the standard hard-sphere resistances that describe pairwise hydrodynamic interactions away from the near-contact region. The results show that permeability removes the contact singularity for non-shearing particle motions, allowing rolling without slip and finite separation velocities between touching particles. Axisymmetric and transverse mobility functions are presented that describe relative particle motion under the action of prescribed forces and in linear flows. At contact, the axisymmetric mobility under the action of oppositely directed forces is$$U/U_0=d_0K^{2/5}$$, where$$U$$is the relative velocity,$$U_0$$is the velocity in the absence of hydrodynamic interactions and$$d_0=1.332$$. Under the action of a constant tangential force, a particle in contact with a permeable half-space rolls without slipping with velocity$$U/U_0=d_1(d_2+\log K^{-1})^{-1}$$, where$$d_1=3.125$$and$$d_2=6.666$$; in shear flow, the same expression holds with$$d_1=7.280$$.more » « less
-
Abstract Letfbe an$$L^2$$-normalized holomorphic newform of weightkon$$\Gamma _0(N) \backslash \mathbb {H}$$withNsquarefree or, more generally, on any hyperbolic surface$$\Gamma \backslash \mathbb {H}$$attached to an Eichler order of squarefree level in an indefinite quaternion algebra over$$\mathbb {Q}$$. Denote byVthe hyperbolic volume of said surface. We prove the sup-norm estimate$$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$ with absolute implied constant. For a cuspidal Maaß newform$$\varphi $$of eigenvalue$$\lambda $$on such a surface, we prove that$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$ We establish analogous estimates in the setting of definite quaternion algebras.more » « less
-
Abstract What proportion of integers$$n \leq N$$may be expressed as$$x^2 + dy^2$$for some$$d \leq \Delta $$, with$$x,y$$integers? Writing$$\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$$for some$$\alpha \in (-\infty , \infty )$$, we show that the answer is$$\Phi (\alpha ) + o(1)$$, where$$\Phi $$is the Gaussian distribution function$$\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$$. A consequence of this is a phase transition: Almost none of the integers$$n \leq N$$can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 - \varepsilon }$$, but almost all of them can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$$.more » « less
-
Well-resolved direct numerical simulations (DNS) have been performed of the flow in a smooth circular pipe of radius$$R$$and axial length$$10{\rm \pi} R$$at friction Reynolds numbers up to$$Re_\tau =5200$$using the pseudo-spectral code OPENPIPEFLOW. Various turbulence statistics are documented and compared with other DNS and experimental data in pipes as well as channels. Small but distinct differences between various datasets are identified. The friction factor$$\lambda$$overshoots by$$2\,\%$$and undershoots by$$0.6\,\%$$the Prandtl friction law at low and high$$Re$$ranges, respectively. In addition,$$\lambda$$in our results is slightly higher than in Pirozzoliet al.(J. Fluid Mech., vol. 926, 2021, A28), but matches well the experiments in Furuichiet al.(Phys. Fluids, vol. 27, issue 9, 2015, 095108). The log-law indicator function, which is nearly indistinguishable between pipe and channel up to$$y^+=250$$, has not yet developed a plateau farther away from the wall in the pipes even for the$$Re_\tau =5200$$cases. The wall shear stress fluctuations and the inner peak of the axial turbulence intensity – which grow monotonically with$$Re_\tau$$– are lower in the pipe than in the channel, but the difference decreases with increasing$$Re_\tau$$. While the wall value is slightly lower in the channel than in the pipe at the same$$Re_\tau$$, the inner peak of the pressure fluctuation shows negligible differences between them. The Reynolds number scaling of all these quantities agrees with both the logarithmic and defect-power laws if the coefficients are properly chosen. The one-dimensional spectrum of the axial velocity fluctuation exhibits a$$k^{-1}$$dependence at an intermediate distance from the wall – also seen in the channel. In summary, these high-fidelity data enable us to provide better insights into the flow physics in the pipes as well as the similarity/difference among different types of wall turbulence.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2023.662
